WLC is Wrong: The Argument for Actual Infinities

[GoldPhoenix]

[N]ow before you can add another particale and do your shuffle, the question should be asked.
"Did you use a complete list of numbers". IE, in the infinite series were all the numbers used.
Of course the answer is yes.

So, then when the Particles move, did you create any new numbers?
The answer is no, of course.

So movement doesn't create new numbers, and all the numbers are used. (IE the list is full), how do you manage to find an unused number?
Or How is the list both sufficient and insufficient to fill the numbers at the same time and in the same sense? Why should we accept such an absurdity as a realizable actuality?

Your argument here is: "The statement 'This set is full' means that you can't add more members without changing the size of the set." Let's call that definition/notion of being full as being "A-full." The answer to your question regarding the nature of the A-fullness of infinite sets is: Not applicable. You're committing a category error; A-fullness does not apply to infinite sets. Infinite sets fail to obey that property, but no one here is disputing that. The question is what is actually broken by a failure of A-fullness to hold for a set of physical objects. You need to either:

A.) Explain to me which current physical law is broken by not accepting A-fullness. In other words, if we break A-fullness as a property for collections of physical objects, then you're alleging that a law of nature has been broken.

B.) Explain to me why A-fullness must itself be considered a physical law. In other words, a set which violates A-fullness doesn't break an existing law, but the requirement that A-fullness apply to all sets itself should be a physical law.

But what you can't do is simply assert by fiat that all sets of physical objects must obey A-fullness, or that a violation of A-fullness is "inconsistent." Inconsistent with what? The laws of logic? Clearly not, even WLC agrees that infinite sets are logically consistent. With the laws of nature? Okay, but which law of nature specifically, and how does the violation of it occur? If not either of those two, then what specifically is being violated by the failure of A-fullness to hold for sets of physical objects?

---Second Supporting Objection---
If You subtracted half the Particles, would you have any less particles?
The answer is no, you would have the same number (IE infinite)

So you are proposing that I take as a realizable reality, that subtraction doesn't change the total.
As WLC says, just don't ask the lady who changes the beds.


Summary of support objections, there is a bit of mathematical slight of hand occurring here. You introduce a mathematical concept that in the end negates some basic mathematical principles (Like subtracting decreases totals) and saying it is logically plausible.


---
Summary of Larger point
Now the above is to show two objections raised by WLC, that are not addressed in the OP. This is why Squatch brings it up immediately. Your conclusion that "No physical contradictions occur" Is not reached by addressing the Objections WLC brings up, it is reached by being silent on them.

This argument is predicated on the same assumptions as the previous one, so see above.

[CliveStaples]

You introduce a mathematical concept that in the end negates some basic mathematical principles (Like subtracting decreases totals) and saying it is logically plausible.

I mean, it is logically plausible. It's how you get the cardinals and the ordinals and so forth. Are you suggesting that transfinite arithmetic is illogical because it lacks well-defined subtraction? "Subtraction decreases totals" supposes that subtraction even makes sense to talk about. Heck, "subtraction decreases totals" doesn't even make sense in the complex numbers, since there's no definable total order on the complex numbers that respects the field operations.

"Subtraction decreases totals" usually means something like this:

If x = y, then x - c < y when c > 0.

Technically, when you subtract negatives, you're adding, e.g. 3 - (-1) > 3, so subtracting -1 from 3 doesn't result in a decrease.

The problem in the complex numbers is that statements like "x > y" aren't really possible without breaking how ">" behaves with respect to addition and multiplication.

You could reformulate your "subtraction decreases totals" to be something like this:

If |x| = |y|, then |x - c| < |x| when |c| > 0

And since complex numbers have a modulus operator (similar to the absolute value operator, it measures "Euclidean distance from zero"). The problem is that this property fails, e.g.

x = i, y = 1, c = i-1:

(i) |x| = |y|, since |x| = |i| = 1 = |1| = |y|
(ii) |c| = |i-1| > 0
(ii) |x - c| = |i - (i-1)| = |1| = 1

So |x-c| = |x|, even though |c| > 0.

---

Your argument relies on "subtraction decreases totals" being such an important property of a system that breaking this property means we should reject the system as being illogical. However, I don't see any definition of "subtraction decreases totals" that isn't immediately broken by very familiar and useful systems like the complex numbers or the natural numbers. Can you give a precise definition of what you mean by "subtraction decreases totals"?

[MindTrap028]

Your argument here is: "The statement 'This set is full' means that you can't add more members without changing the size of the set

First of all, I don't think "size of set" is being addressed here, because Infinite is being rejected as a real thing.

Full here means that no further accommodations can be made.

So your right, Infinites can not be full.. but hotels can.
What is broken is the connection between and Idea, and reality.
This is evidenced by WLC's statment, "No vacancies, rooms available".

IE an contradiction.


Second, I'm not certain that was the premise I offered.

For example I asked this question which you did not answer "How is the list both sufficient and insufficient to fill the numbers at the same time and in the same sense?"
I also spoke to "all the numbers being used". Which I don't think you addressed iether.


You may be correct that infinite as a concept doesn't have the idea of "full" applied to it, But hotels do. That idea is more to do with sufficient than the size.

A.) Explain to me which current physical law is broken by not accepting A-fullness. In other words, if we break A-fullness as a property for collections of physical objects, then you're alleging that a law of nature has been broken.

Well, first lets get one thing clear. I was offering my attempt at repeating an objection which you had not addressed in the OP.
so, the idea of "my" objection or claim is a loose one at best.

As to what law of nature you have broken, the claim is that you haven't presented something that is coherent to begine with. You couldn't possibly point to what physical law a married bachelor contradicts.
Because I haven't pointed to a coherent and meaningful thing.

-Summary--
I think you addressed a false premise that I didn't offer (though It may be a valid line in itself), and as such you didn't address my objections, or my question.
In so doing you shifted the burden to me when the burden lay on you to first make a statement of meaning before demanding some physical violation.


-----

Your argument relies on "subtraction decreases totals" being such an important property of a system that breaking this property means we should reject the system as being illogical. However, I don't see any definition of "subtraction decreases totals" that isn't immediately broken by very familiar and useful systems like the complex numbers or the natural numbers. Can you give a precise definition of what you mean by "subtraction decreases totals"?

First of all, I want to thank you for your post because you did a good job of explaining the concept, and trying to apply it in a superior form, and then rebutting that as well.

O.k. that being said I did not do a good job of identifying or explaining my objection regarding subtraction.
Specifically the idea of "decreasing totals" was not the correct terminology to use.

What I mean, is that when you remove A, A is removed.
If you remove A, and A remains, then the statement that you have removed A is nonsensical.

The idea that A can be infinite, A is then removed and A remains is the basis for the objection regarding subtraction.

While such reasoning is allowed by the idea of inifite, there isn't very good reason to reject the former reasoning for it or accept infinite as a reasonable exception.

You would probably do better to review WLC's objection that I am trying to convey...



------------------------------------


TO GP again

I believe my larger point is still un-addressed.

Summary of Larger point
Now the above is to show two objections raised by WLC, that are not addressed in the OP. This is why Squatch brings it up immediately. Your conclusion that "No physical contradictions occur" Is not reached by addressing the Objections WLC brings up, it is reached by being silent on them.

As to a new point, the OP asks, and you ask in the thread many times for a physical objection for why Infinite can't exist. Is there any problem with offering a logical objection?
I mean, suppose my objection that your talking about a married bachelor is valid, does it some how fail because it isn't based on physics?

[CliveStaples]

If you remove A, and A remains, then the statement that you have removed A is nonsensical.

This is an excellent intuition, I think, but it stands for infinite sets. If you have S = {0,1,2,...}, and you take away all of S, you end up with the empty set {} (S - S = {}). If you take away portions of S, you're left with the remainder; e.g. S - {1,2,3,...} = {0}.

What fails for infinite sets is not that you can take away the whole set and still have something left, but rather that there are portions of S that contain just as many objects as S. That is, {1,2,3,4,...} contains just as many objects as {0,1,2,3,4,...}.

For finite sets, every portion contains fewer objects than the whole. So if you have a finite set S = {1,2,3, ... , n}, and you take away a portion of size m, you're always left with a remainder of size n-m no matter which m-sized portion you chose. So you have a cardinality-based shortcut to find the cardinality of the remainder.

Whereas for infinite sets, there are lots of infinity-sized portions of S. The remainder, as with finite sets, is determined by what's contained in the portion you remove. But unlike finite sets, there's no cardinality-based shortcut to find the cardinality of the remainder. And while it would be nice for the cardinality of the remainder to always be strictly based on the cardinality of the portion you remove, sadly that shortcut is not always available. (This result is certainly counter-intuitive at first, but it's easier to swallow when you remember that the remainder is always strictly determined by what's contained in the portion you remove, even for infinite sets.)

[MindTrap028]

What fails for infinite sets is not that you can take away the whole set and still have something left, but rather that there are portions of S that contain just as many objects as S. That is, {1,2,3,4,...} contains just as many objects as {0,1,2,3,4,...}.

First, thanks for your whole post.

Also, the idea of a portion the same size as the whole is contradictory for an actual object. Ideas can operate that way, but not reality.

(This result is certainly counter-intuitive at first, but it's easier to swallow when you remember that the remainder is always strictly determined by what's contained in the portion you remove, even for infinite sets.)

I'm afraid I don't think that is born out for infinite sets.
Rather, the remainder appears to be determined by the "where" of what you remove. Remove an infinite number of items above, "5" and you have a remainder of 5.
Or
Suppose you remove every other one, you have removed half of the total.. that is equal to the whole, yet identical to the number that left only 5. That is not simply intuitively incorrect, it is illogical.
As I pointed out, the sufficiency and insufficiency of the list, at the same time and in the same sense. .. that is the definition of the violation of the logical law of identity.


Infinities seem to treat location as relevant to sufficiency(or end size), as well as introducing a directional bias.

Move all guests up one room and you have a vacant room.. move them all back one room.. and no more vacancies. It treats motion as relevant to sufficiency to fill the rooms.
Is it going out on a limb to say that such a thing is specifically and observably not how reality works?

All of the objections don't simply rest on the counter intuitive nature, but the absurdity and logical incoherence of some of the results.
Infinity is fine for mind games, but seems totally implausible for any sort of logically possible world.

[CliveStaples]

First, thanks for your whole post.

Also, the idea of a portion the same size as the whole is contradictory for an actual object. Ideas can operate that way, but not reality.

You need to support this assertion.

What physical law would it contradict for a countably-infinite set of particles to have a countably-infinite subset of particles?

I'm afraid I don't think that is born out for infinite sets.
Rather, the remainder appears to be determined by the "where" of what you remove. Remove an infinite number of items above, "5" and you have a remainder of 5.

The remainder--{1,2,3,4,5}--is strictly determined by the contents of the portion you removed--{6,7,8,9,...}. So it seems that the remainder in this case was strictly determined by the contents of the portion that was removed.

Or

Suppose you remove every other one, you have removed half of the total.. that is equal to the whole, yet identical to the number that left only 5. That is not simply intuitively incorrect, it is illogical.

You removed a different portion and got a different remainder. Here, you removed {2,4,6,8,...} and were left with {1,3,5,7,...} as the remainder. So again it seems that the remainder was strictly determined by the contents of the portion that was removed.

You didn't remove the same portion. You removed a different portion that contained the same number of objects and got a different remainder. For this to be a logical contradiction, you would need the following principle to be a logical necessity:

(P) If two portions of a whole contain the same number of objects, then their remainders will contain the same number of objects.

This principle, P, seems intuitively compelling, but as you can see in the case of infinite sets, it doesn't always hold.

But you don't just claim that the result is unintuitive; you claim that it is illogical. That is, your claim isn't merely that P is intuitively compelling, but that it is irrational to deny and is logically necessary.

Your argument, therefore, is in need of support for the claim that P is logically necessary. A logical proof would suffice.

As I pointed out, the sufficiency and insufficiency of the list, at the same time and in the same sense. .. that is the definition of the violation of the logical law of identity.

I have no idea what you're talking about. What does it mean for a list to be "sufficient" or "insufficient"?

Infinities seem to treat location as relevant to sufficiency(or end size), as well as introducing a directional bias.

I'm not sure what you mean by "location". Perhaps you mean something like this:

Let S = {s1,s2,...} be an infinite subset of N = {0,1,2,3,...} and remove it. Call this remainder R (so R = {r1, r2, ...} and is possibly finite). Let Sk denote a shift of S by k units, i.e. Sk = {s1+k, s2+k, s3+k, ...}. Let Rk denote the remainder when Sk is removed (so Rk = N - Sk). In particular, S = S0 and R = R0.

Now, I'm not sure what it means to "treat location as relevant to sufficiency" or what "direction bias" means, but perhaps I can make a close guess.

We'll say that location is relevant to sufficiency if |Rk| depends on the value of k--scoot the entries of S over different amounts and you get different-sized remainders.

A few points:

EDIT: The proof in (1) has a fatal error. (2) is unaffected by this error. The proof in (1) relies on the existence of additive inverses and is thus not applicable to N. (It is, however, applicable to Z = {..., -1, 0 , 1, ...}.) In fact, a counterexample in N can be given. Take S = n, k=1. Then N-S = {} but N-(S+1) = {0}.

This raises an interesting point, however: it isn't infinite sets that are objectionable to due to location being relevant to sufficiency, since this objection fails for Z, which is an infinite set. The difference between Z and N isn't cardinality, but rather order type and algebraic structure. It is N's order type and algebraic structure that determine whether location is relevant to sufficiency, not its cardinality. It would be trivial to amend the order structure of the particles to be isomorphic to Z rather than N.

It would be strange, I think, to accept Z as reasonable (since location isn't relevant to sufficiency in Z) but reject N, a subset of Z, as being unreasonable.

(1) Location isn't relevant to sufficiency.

Proof:

To prove this, I'll need the following lemma:

Let R+k denote the set {r1 + k, r2 + k, ...}.

(LEMMA) Rk = R+k = {r1 + k, r2 + k, ...}.

(R+k ⊆ Rk)
Suppose x \in R+k. Then x = rn+k for some n.
Suppose x \in Sk. Then x = sm+k for some m. But then rn + k = sm + k, so rn = sm. Thus rn is in S. This is a contradiction, since rn \in R and R = N - S.
Thus x \notin Sk. So x \in N - Sk = Rk.
QED.

(Rk ⊆ R+k)
Let x \in Rk.
Suppose x-k is in S. Then x-k = sn for some n. Thus x = sn+k. Hence x \in Sk. This is a contradiction, since x \in Rk and Rk = N - Sk.
Thus x-k \in N - S. So x-k \in R. Thus x-k = rm for some m. Thus x = rm + k. Hence x \in R+k.
QED.

Thus Rk = R+k.
QED.

(THEOREM) |Rk| = |R| for all k in N.

Proof:

Let k be any element of N = {0,1,2,...}. Define f:Rk \to R by

f(rn+k) = rn

By (LEMMA), Rk = R+k so every element of Rk can be written as rn+k for some n. Thus f is indeed a map from Rk to R.

f is well-defined:
rn+k = rm + k implies rn = rm and hence f(rn+k) = f(rm+k).

f is injective:
f(rn+k) = f(rm+k) implies rn = rm which implies rn+k = rm+k.

f is surjective:
Let rm \in R. Then rm+k \in R+k = Rk, and f(rm+k) = rm.

Thus f is a bijection.
QED.

By (THEOREM), |Rk| = |R| = |Rj| for all k,j. Thus location is not relevant to sufficiency.

(2) What's wrong with location being relevant to sufficiency? What theorem or axiom of logic does this break?

Move all guests up one room and you have a vacant room.. move them all back one room.. and no more vacancies. It treats motion as relevant to sufficiency to fill the rooms.
Is it going out on a limb to say that such a thing is specifically and observably not how reality works?

I don't know what it means for motion to be "relevant to sufficiency to fill the rooms", or what law of logic or physics this would break.

All of the objections don't simply rest on the counter intuitive nature, but the absurdity and logical incoherence of some of the results.

If this is so, then you should be able to name what logical axiom or theorem is broken by the results.

[GoldPhoenix]

I'm just going to address your overall question MT, which seems to succinctly be stated at the end of your post:

As to a new point, the OP asks, and you ask in the thread many times for a physical objection for why Infinite can't exist. Is there any problem with offering a logical objection?
I mean, suppose my objection that your talking about a married bachelor is valid, does it some how fail because it isn't based on physics?

Yes, there is. At least if we wanted to address and attack/defend WLC's point. To quote Squatch's post #3, which contains an excerpt of WLC's thoughts on your particular line of questioning:

"At this point, we might find it profitable to consider several objections that might be raised against the argument [that actual infinities are impossible]. First let us consider objections to (2.11). Wallace Matson objects that the premiss must mean that an actually infinite number of things is logically impossible; but it is easy to show that such a collection is logically possible. For example, the series of negative numbers {. . . -3, -2, -1} is an actually infinite collection with no first member. Matson's error here lies in thinking that (2.11) means to assert the logical impossibility of an actually infinite number of things. What the premiss expresses is the real or factual impossibility of an actual infinite. To illustrate the difference between real and logical possibility: there is no logical impossibility in something's coming to exist without a cause, but such a circumstance may well be really or metaphysically impossible. In the same way, (2.11) asserts that the absurdities entailed in the real existence of an actual infinite show that such an existence is metaphysically impossible. Hence, one could grant that in the conceptual realm of mathematics one can, given certain conventions and axioms, speak consistently about infinite sets of numbers, but this in no way implies that an actually infinite number of things is really possible."

In other words, WLC is aware of modern mathematics and the copious use that infinity gets (precisely because it is logically consistent, as WLC himself attests above), thus he is not making the claim (again, due to trivial counter-examples like the one he gave) that infinity is logically inconsistent.


So if you want to argue that infinity itself is illogical, then that's fine, but you're now making a stronger claim than WLC is (and precisely because WLC knows that such a strong claim is not a defensible position). And thus you would need to defend that argument with a much more formal argument than you're currently giving (i.e. precisely and thoroughly spell out the contradiction that you think exists). Clive seems to be doing this with you now, but I thought I'd make this clarification regarding WLC's own positions.

---

A side statement: Of course, this gets the heart of what makes WLC's argument fallacious. WLC expects you not to understand advanced mathematics (he certainly doesn't), so when he presents this "ridiculous" example of Hilbert's Hotel, WLC then sets up his audience to have a host of confusions and equivocations. Are they objecting logically to infinity? No, that trivially fails, Hilbert himself pushed the work of ZFC for how to consistently deal with infinite sets. Are they objecting to the metaphysics of infinity? Well, not even that, clearly, unless you want to give up Platonist perspective of mathematics and logic coming from god --something I'm okay with, but very few Christians (and pure mathematicians) I've met are. So, probably they don't want that, either. Are they objecting to infinities based on the laws of Nature? Well, on this thread, it doesn't really seem so. The whole point boils down to simple confusions in one's intuition. If you resolve the errors lying in your intuition, then the entire contradiction (logically, metaphysically, and nomologically) disappears wholesale. There is no logical contradiction, there can be assigned logically self-consistent systems that makes sense of infinite sets --thus the metaphysical argument is dubious at best; if you're a non-Intuitionist Platonist then it's necessarily false-- and there's clearly no physical law to appeal to regarding the falsity of actual infinities --as I've shown, i.e. in quantum mechanics, only the converse appears to be true.

So to my mind, being able to understand precisely how mathematical infinities are logically possible is what dispels all of the other problems.

[MindTrap028]

You need to support this assertion.
What physical law would it contradict for a countably-infinite set of particles to have a countably-infinite subset of particles?

Well, I'm not really certain what name it would go by, but I think I can illustrate it.

-Fish example-
In the OP a meter as the basis for it's example. That is the length of a nice size fish. If I wish to give you a whole fish, and keep one for myself, the OP implies that all I need to do is chop it up into small enough pieces, then I can give you a whole fish, without decreasing the amount of fish I have. (ask if you need more explanation)

The remainder--{1,2,3,4,5}--is strictly determined by the contents of the portion you removed--{6,7,8,9,...}. So it seems that the remainder in this case was strictly determined by the contents of the portion that was removed.

I was talking about the amount of the remainder, depends on the location of the amount removed.
Here, I guess I was speaking to the sum of the remainder, and not the objects in the remainder.

I was not aware that math was effected by the location.

You removed a different portion and got a different remainder.

See above, same sum.


---

This principle, P, seems intuitively compelling, but as you can see in the case of infinite sets, it doesn't always hold.

But you don't just claim that the result is unintuitive; you claim that it is illogical. That is, your claim isn't merely that P is intuitively compelling, but that it is irrational to deny and is logically necessary.

Your argument, therefore, is in need of support for the claim that P is logically necessary. A logical proof would suffice.

Fair enough, and I will endeavor to offer that here.
I am under the assumption that math is the same every time you do it correctly, is that a safe assumption?
so, if you take 1 and you add another 1, then you will always get 2
Also, when you have 1 and you remove 1, then you will always get 0.
This process is irrelevant to who does it, or where it is done, or the order of the removal(in the case of multiple elements).

This is of course flowing from the logical law of identity. (correct/agree?) If not feel free to offer a reading reference for the logical basis of mathematics.

So, if you have Infinity, and you subtract 1, you will always get the same answer.
But this is not true when you subtract infinity. Suddenly an irrelevant element to mathematics, becomes relevant, Namely the location of the subtraction.

This is evident by the fact that if you have infinity and you subtract infinity you can get any and all possible sums, based on what the location is of that subtraction. (though all the while subtracting the same amount.. from the same amount, IE in the same way and in the same sense.)
This is evidence that infinity violates the law of identity. It is both itself and not itself at the same time and in the same sense.

Question, is there any other way to disprove math than to get varying answer from the same problem (not problems with a variable, but with actualized knowns)?
Should we accept that location of subtraction is relevant to the expected sum for all of mathematics, if not why should we accept infinite as the exception as a valid reason for taking it as reasonable?
I mean, if I said 5-4=10, and you object(on whatever lines) and my answer was.. Yes but the number 5 is the exception to the general rules of mathematics.

(*note* you have done well in this thread clive to see past my poor wording and have made honest attempts to offer a superior version on several occasions. I welcome any corrections here, as this is my personal attempt)

Question, If it doesn't violate the law of identity... what exactly is infinity?

-------

I have no idea what you're talking about. What does it mean for a list to be "sufficient" or "insufficient"?

Sufficiency is it's ability of one list to fill the other.
So, with the hotel, if you start off at room 1, then the list of guests is sufficient to fill the hotel.
If however you start at room (whatever) the list is not sufficient (as the list is completely used with no remainder, and yet there are rooms available).

Now, I'm not sure what it means to "treat location as relevant to sufficiency" or what "direction bias" means, but perhaps I can make a close guess.

What I mean by direction bias, is that in the hotel if you move all the guests one direction, you have an extra room(s).. where as if you move it the other direction then you have extra guest(s).
This is evidence that the direction of the math is relevant to the end sum.

Now, I'm not sure what it means to "treat location as relevant to sufficiency" or what "direction bias" means, but perhaps I can make a close guess.

We'll say that location is relevant to sufficiency if |Rk| depends on the value of k--scoot the entries of S over different amounts and you get different-sized remainders.

sincerely thanks for all that followed this quoted portion in your post. Unfortunately, I'm just too stupid to understand it. My brain simply doesn't process information that way, you may as well be speaking Greek.
Of course I welcome you to make any attempts at translating it with me.. but I don't think you really have that burden to do so.
Of course my failure to understand what you said, doesn't mean in anyway that you have not responded.. or even debunked what I have said here. I simply am not equipped at his point to know. I am very sorry for this shortcoming, and will accept any reference to this section as the final word in response to what I have said here. So.. while I continue, it isn't out of ignoring this section because it is to inconvenient.. I just don't get it.

To that end, I want to refer you to what I tell GP on the matter of communication and the challenge presented by responding to WLC's position. If he is persuasive only because he is able to communicate to people like me, it is your challenge to find an equally clear way to rebut it. As GP says, "being able to understand precisely how mathematical infinities are logically possible is what dispels all of the other problems. "

I think I can get the concepts.. just not in a super technical text to present it.

(2) What's wrong with location being relevant to sufficiency? What theorem or axiom of logic does this break?

I would say that it seems to be a categorical error. Location is categorically not relevant to math problems.
Unless you suspect that somewhere in the universe 1+1=3 is a valid possibility.

I don't know what it means for motion to be "relevant to sufficiency to fill the rooms", or what law of logic or physics this would break.

I think I explained this above.

If this is so, then you should be able to name what logical axiom or theorem is broken by the results.

I also think I did this above.

---------------------------

So if you want to argue that infinity itself is illogical, then that's fine, but you're now making a stronger claim than WLC is (and precisely because WLC knows that such a strong claim is not a defensible position). And thus you would need to defend that argument with a much more formal argument than you're currently giving (i.e. precisely and thoroughly spell out the contradiction that you think exists). Clive seems to be doing this with you now, but I thought I'd make this clarification regarding WLC's own positions.

First, I have no real desire to go beyond what WLC has laid out, because that doesn't support my greater point that I made earlier.
Secondly, I think I am staying consistent with the context of why it can't be a reality.

A side statement: Of course, this gets the heart of what makes WLC's argument fallacious. WLC expects you not to understand advanced mathematics (he certainly doesn't), so when he presents this "ridiculous" example of Hilbert's Hotel, WLC then sets up his audience to have a host of confusions and equivocations. Are they objecting logically to infinity? No, that trivially fails, Hilbert himself pushed the work of ZFC for how to consistently deal with infinite sets. Are they objecting to the metaphysics of infinity? Well, not even that, clearly, unless you want to give up Platonist perspective of mathematics and logic coming from god --something I'm okay with, but very few Christians (and pure mathematicians) I've met are. So, probably they don't want that, either. Are they objecting to infinities based on the laws of Nature? Well, on this thread, it doesn't really seem so. The whole point boils down to simple confusions in one's intuition. If you resolve the errors lying in your intuition, then the entire contradiction (logically, metaphysically, and nomologically) disappears wholesale. There is no logical contradiction, there can be assigned logically self-consistent systems that makes sense of infinite sets --thus the metaphysical argument is dubious at best; if you're a non-Intuitionist Platonist then it's necessarily false-- and there's clearly no physical law to appeal to regarding the falsity of actual infinities --as I've shown, i.e. in quantum mechanics, only the converse appears to be true.

So to my mind, being able to understand precisely how mathematical infinities are logically possible is what dispels all of the other problems.

I am very well aware that this is not my area of expertise, and as I noted to Clive I am a bit unwieldy with the terms and mathematical presentation. This is not to say that you guys have hammered that point, or grandstanded inappropriately. That said, take note of my exchange with clive in regards to his technical response.

But I would have to say that the challenge for the OP, if it wants to counter WLC, is that it must find a way to effectively reach the audience such as WLC does.


As a lay person, what you have told me is that If I wanted to share a fish with you and Clive then I can give you each a 3rd by cutting it into 3 pieces.
but, if I'm creative enough and manage to cut it into infinity small pieces, then You, Clive, myself and the whole world can be given not a portion of a fish, but a whole fish.

Now, the fool that I am I may not be able adequately describe in mathematical, physical or even logical terms why that isn't true.. but dude... You ain't Jesus *J*(pun on feeding the five thousand)


Joking aside, let me summarize some observations about the way infinite are used here.. and you can let me know if the observations are correct.

1) Infinite introduce a directional bias in regards to outcome. (IE the direction of movement of the tenants dictates if the hotel is full, overflowing with guests, or has rooms to spare).
2) Infinite introduce a location bias in regards to outcome. (IE the location of the division determines if it's possible to accomplish. Such that if you tried to cut an infinite hotel in half.. you could not find the center.. but if you take every other room out you have accomplished the task)
3) Infinite tell me that If I desire to give you a fish, I need only cut it small enough to feed the world. That one fish = infinity fish. (IE destroying what it means to be 1)

[CliveStaples]

Well, I'm not really certain what name it would go by, but I think I can illustrate it.

-Fish example-
In the OP a meter as the basis for it's example. That is the length of a nice size fish. If I wish to give you a whole fish, and keep one for myself, the OP implies that all I need to do is chop it up into small enough pieces, then I can give you a whole fish, without decreasing the amount of fish I have. (ask if you need more explanation)

Are you trying to say there's a conservation of mass issue?

I was talking about the amount of the remainder, depends on the location of the amount removed.
Here, I guess I was speaking to the sum of the remainder, and not the objects in the remainder.

I take it "the location of the amount removed" just means "which particular objects you choose to remove".

I was not aware that math was effected by the location.

Sometimes properties/computations are affected by location, sometimes they aren't. For instance, in probability and statistics, if you're using the uniform distribution then your probabilities depend only on the length of the interval, not its position; no other probability distribution has this feature.

What contradiction, if any, are you trying to identify? It's true that arithmetic on the cardinals is not isomorphic/identical to arithmetic on the real numbers. But there's plenty of algebraic objects whose operations aren't isomorphic/identical to the reals.

Fair enough, and I will endeavor to offer that here.
I am under the assumption that math is the same every time you do it correctly, is that a safe assumption?

Generally speaking, yes, but it depends on what you mean by "math". If you're taking samples from a random distribution, you might get different values each time, for example.

so, if you take 1 and you add another 1, then you will always get 2
Also, when you have 1 and you remove 1, then you will always get 0.
This process is irrelevant to who does it, or where it is done, or the order of the removal(in the case of multiple elements).

In the real numbers, order doesn't matter, since multiplication and addition are both associative and commutative. There are many algebraic structures that are either not associative, not commutative, or neither. In general, order does matter; associative and commutative structures are particularly well-behaved.

So the answer is that it depends on the algebraic structure you're working in.

This is of course flowing from the logical law of identity. (correct/agree?) If not feel free to offer a reading reference for the logical basis of mathematics.

No, x=x doesn't imply x+y = y+x. If you're doing subtraction in the reals, x-(y-z) =/= (x-y)-z, so it matters in what order you do your subtractions--i.e., do you subtract z from y, then subtract that result from x, or do you subtract y from x, and then subtract z from that result? But if you're doing addition, x+(y+z) = (x+y)+z, so order doesn't matter--i.e., you can add y to z, then add that to x, or you can add y to x, then add that to z.

Matrix multiplication is associative, so A(BC) = (AB)C, but not commutative, so AB does not always equal BA.

For an operation that is commutative but not associative, consider the set S={Rock, Paper, Scissors}, and define multiplication as xy = the winner of x and y. So (Rock)(Paper) = Paper, (Rock)(Scissors) = Rock, etc. This operation is not associative, since [(Rock)(Paper)](Scissors) = (Paper)(Scissors) = Scissors, but (Rock)[(Paper)(Scissors)] = (Rock)(Scissors) = Rock. It is, however, commutative, since the winner of x and y is the winner of y and x.

So, if you have Infinity, and you subtract 1, you will always get the same answer.

Sure, if you're going with cardinals so your operations are defined in terms of set operations.

But this is not true when you subtract infinity. Suddenly an irrelevant element to mathematics, becomes relevant, Namely the location of the subtraction.

Well, you're using "location" in two different senses, here. One is algebraic, i.e. the location of "-" or the order of operations. The other is set-theoretic, e.g. which infinite subset of {0,1,...} you choose to remove. Both are relevant to the computations involved. So I don't know why you think that "location" is irrelevant to mathematics.

This is evident by the fact that if you have infinity and you subtract infinity you can get any and all possible sums, based on what the location is of that subtraction. (though all the while subtracting the same amount.. from the same amount, IE in the same way and in the same sense.)

Well, not in the same way, actually. You can subtract infinity from infinity in lots of different ways, if you're talking about taking relative complements of infinite subsets. You could take the evens out of the naturals, or the primes out of the naturals, etc.

The operations here are defined in terms of sets and cardinalities, which means certain formal expressions (like aleph0 - aleph0) are left undefined. This is not a contradiction, and follows (as you know) from the definition of arithmetic on infinite cardinals.

This is evidence that infinity violates the law of identity. It is both itself and not itself at the same time and in the same sense.

This is manifestly false. Each cardinal is equal to itself.

Question, is there any other way to disprove math than to get varying answer from the same problem (not problems with a variable, but with actualized knowns)?

Well, this is why they leave subtraction undefined except in certain specific expressions. Transfinite arithmetic is defined in terms of set cardinalities, and (as you've seen) |A-B| doesn't just depend on |A| and |B|. So they leave |A|-|B| undefined.

But to answer your question, yes, there are other methods of disproof than showing that a=b and a=c but b =/= c. Generally to disprove something you need to show that it entails a contradiction.

Should we accept that location of subtraction is relevant to the expected sum for all of mathematics, if not why should we accept infinite as the exception as a valid reason for taking it as reasonable?

Well, there's lots of different operations you could define on the class of cardinals, or the extended reals, or the hyperreals, or whatever you like. You could call any of them "subtraction", if you wish.

But regardless of what operations you want to define, the question at hand is how infinite sets behave. And it seems clear that you have the following:

|{0,1,2,...} - {1,2,...}| = |{0}| = 1
|{0,1,2,...} - {5,6,7,...}| = |{0,1,2,3,4}| = 5
|{0,1,2,...} -{0,2,4,6,...}| = |{1,3,5,7,...}| = |{0,1,2,...}|

So it seems clear that |A-B| doesn't just depend on |A| and |B|.

The arithmetic here is defined in a way that depends on the behavior of the sets involved, i.e. in the cardinals, 2+3=5 because whenever two disjoint sets X and Y are such that |X| = 2 and |Y| = 3, we have |X U Y| = 5. aleph0 - 1 = alpeh0 because whenever |X| = aleph0 and |Y| = 1, |X-Y| = aleph0. aleph1-aleph0 = aleph1, since whenever |X|=aleph1 and |Y| = aleph0, |X-Y| = aleph1.

The problem with expressions like |X| - |X| when your operations are defined in terms of the behavior of the sets involved is that |X-Y| can be any cardinality less than or equal to |X|, even when |Y| = |X|.

But you could come up with different operations if you like. You don't have to base |X|+|Y| on the behavior of |XUY|, or |X|-|Y| on the behavior of |X-Y|.

I mean, if I said 5-4=10, and you object(on whatever lines) and my answer was.. Yes but the number 5 is the exception to the general rules of mathematics.

(*note* you have done well in this thread clive to see past my poor wording and have made honest attempts to offer a superior version on several occasions. I welcome any corrections here, as this is my personal attempt)[/quote]

Well, if I knew that you were using "-", "5", "4", "=", and "10" with their standard definition, I would give you a proof that 5-4 = 1, and 1 =/= 10.

But there's lots of special exceptions in mathematics. x/y is defined--except for y=0, whoops. Every prime is odd--except for 2, whoops. Every integer doesn't have a multiplicative inverse in Z--except for 1 and -1, whoops.

Question, If it doesn't violate the law of identity... what exactly is infinity?

The answer is it depends on the context.

Sometimes "infinity" refers to an element in a particular set called the extended real numbers or the extended real line. This set is basically the real numbers (which can be defined/constructed in a few different but equivalent ways: all finite and infinite decimals, or all cauchy sequences of rationals, or all dedekind cuts of rationals, etc.) along with two elements, infinity and -infinity. There are a few operations defined on this set, which are inherited from the familiar addition and multiplication on the reals.

Sometimes "infinity" refers to cardinals or cardinality. Here you're working in set theory, and you're mainly concerned with sets and one-to-one functions (or bijections) between them. Set operations include union (X∪Y, the set whose elements are elements of X, elements of Y, or both), intersection (X∩Y, whose elements are elements of both X and Y), relative complement (X-Y, whose elements are elements of X but not elements of Y), and powerset (P(X), whose elements are subsets of X).

The most basic definition on cardinals is equality: if there is a one-to-one function from X to Y, then |X| = |Y|. If there's a one-to-one function from X to a subset of Y, then |X| <= |Y|. If there's a one-to-one function from X to a subset of Y but not a one-to-one function from X to Y, then |X| < |Y|. The Cantor-Schroeder-Bernstein theorem establishes that for any sets X,Y, we have |X| = |Y|, |X| < |Y|, or |X| > |Y|. Cantor's theorem establishes that |X| < |P(X)|.

We can group sets according to their cardinality: if there's a one-to-one function from X to {1,2,...,n}, then |X| = n. So 1 = |{0}| = |{1}| = |{Gordon Freeman}| = |{red}| = ...; 2 = |{0,1}| = |{red, blue}| = ...; etc. The cardinals or cardinal numbers refer to these collections, and arithmetic on the cardinals is defined in terms of set operations on the collection of sets in those cardinals.

The cardinalities of infinite sets are usually written as aleph numbers, whose definition is tricky to understand. Aleph0 is easy: it's just the cardinality of the natural numbers. So if there's a bijection from X to the natural numbers, then |X| = aleph0. Sets whose cardinality is aleph0 are called countably infinte, and countable sets are either finite or countably infinite.

Larger aleph numbers are defined in terms of ordinals or ordinal numbers, which are a lot tougher to explain. (Basically the ordinals capture order properties as well as cardinality properties.) Aleph1 is equal to the cardinality of the set of all countable ordinals.

Sufficiency is it's ability of one list to fill the other.
So, with the hotel, if you start off at room 1, then the list of guests is sufficient to fill the hotel.
If however you start at room (whatever) the list is not sufficient (as the list is completely used with no remainder, and yet there are rooms available).

This is because there is a one-to-one function between any two countably infinite sets, and the set of all rooms whose number is greater than n is countably infinite.

What I mean by direction bias, is that in the hotel if you move all the guests one direction, you have an extra room(s).. where as if you move it the other direction then you have extra guest(s).
This is evidence that the direction of the math is relevant to the end sum.

I don't understand what you're trying to identify as problematic, here.

If you have a countably infinite list, you can leave off any finite portion of the list and still have countably infinite members on the list, which you can then assign to rooms using a one-to-one function since there are countably infinite rooms.

"Moving in one direction" means assigning the first guest starting at the nth room, and "moving in the other direction" means assigning the first room starting at the nth guest. As you would expect, "moving in one direction" leaves off rooms 1 through n-1, and "moving in the other direction" means leaving off guests 1 through (n-1).

I'm not sure how this means that "the direction of the math is relevant to the end sum" in any kind of problematic fashion. If you leave off rooms, you have extra rooms. If you leave off guests, you have extra guests.

sincerely thanks for all that followed this quoted portion in your post. Unfortunately, I'm just too stupid to understand it. My brain simply doesn't process information that way, you may as well be speaking Greek.
Of course I welcome you to make any attempts at translating it with me.. but I don't think you really have that burden to do so.
Of course my failure to understand what you said, doesn't mean in anyway that you have not responded.. or even debunked what I have said here. I simply am not equipped at his point to know. I am very sorry for this shortcoming, and will accept any reference to this section as the final word in response to what I have said here. So.. while I continue, it isn't out of ignoring this section because it is to inconvenient.. I just don't get it.

Basically, sometimes position matters and sometimes it doesn't. It's nice when it doesn't, because it means you have fewer things you need to pay attention to, but in generally it's not logically necessary for location to be irrelevant. (In certain specific cases it is--e.g. arithmetic on the reals, where addition and multiplication are associative and commutative).

To that end, I want to refer you to what I tell GP on the matter of communication and the challenge presented by responding to WLC's position. If he is persuasive only because he is able to communicate to people like me, it is your challenge to find an equally clear way to rebut it. As GP says, "being able to understand precisely how mathematical infinities are logically possible is what dispels all of the other problems. "

I think I can get the concepts.. just not in a super technical text to present it.

Well, the issue is that it's really easy to be wrong about infinities and it's hard to be right about them, which is why we didn't really have a great handle on it until Cantor, and even then his work was rejected by many of his contemporaries.

I would say that it seems to be a categorical error. Location is categorically not relevant to math problems.
Unless you suspect that somewhere in the universe 1+1=3 is a valid possibility.

Not every algebraic object has associative and commutative operations. Location is relevant to determining the probability that a random variable will fall into an interval of a certain length, for example.

[GoldPhoenix]

Okay, MT, I think I understand your intuition and where the problem lies. But I just want us to be super clear by what our words mean. So let me define a few terms here so we can be super clear by what I mean. So please read this introductory discussion, please read my questions to your post, and then respond to my questions. But it's vital that you understand the introductory definitions.

---

Def 1: A set is purely a collection of objects. The only information it contains is the list of unique objects. The empty set, i.e. the set of no elements is just denoted as {}.

Example 1.1: Let's take a set with two elements, let's call them A and B (A could be a dog, B could be a cat, for example), and the set of them we will denote as {A, B}. The following statements are true:

1.) {A, B} = {B, A}. Order doesn't matter, only what is actually contained in the set. In other words, {A,B} contains the exact same unique members as {B,A}; so at the level of pure sets, they're equivalent.

2.) {A,A,B} = {A,B}. Repeating elements is irrelevant, it only matters what the unique members are, so repeating A changes nothing because there's still only two unique members, A and B.

Def 2: Given two sets, X and Y, let the union of X and Y be defined as combining all of the unique elements, and let's denote the union by "X + Y".

Example 2.1: Let X = {1,2} and Y = {3}, then, X + Y = {1,2,3}. It also equals {3,2,1}, {2,3,1}, etc, again because order doesn't matter.

Example 2.2: Let X = {a,b}, Y = {b,c}. Then, X + Y = {a,b,c}. Note that b was repeated, but we needn't write it down twice, since only unique elements of the set matter.

Example 2.3: Let X = {a,b} and let Y = {}. Then X + Y = X.

Def 3: Let X and Y be sets. Let the difference of X minus Y be defined as removing the unique elements of Y from X. We will denote this as "X - Y."

Example 3.1: Let X = {1,2,3}, and let Y = {1,3}. Then X - Y = {2}. Also, we see that Y - X = {}, because Y contains no unique elements that aren't in X.

Def 4: The size of a set, X, is the number of elements contained in the set. For finite sets, this means that the size of the set will be a counting number (e.g. 0, 1, 2, and so on); for infinite sets, we will have to use ordinal numbers. For the purposes of this conversation, let's say that there's an infinity called aleph-0. An infinite set X is said to have cardinality aleph-null if it is the same size as the counting numbers {0, 1, 2, 3, ...}. Note that we do not demand that if X is a subset of Y, but both are infinite, they are still the same size.

Example 4.1: Let X = {a,b,c} and Y = {c,d}. The size of X is 3, the size of Y is 2, the size of X + Y is 4, and the size of X - Y is 2.
Example 4.2: Let the set X = {2,4,6,8, ..., 2n, ...}. The size of X is aleph-null.

That may look like a bit of text, but it's just three points, really: A set is a collection of unique elements with a given size. We may make a new set by adding the unique elements of two previous sets. We may also subtract one set from another by removing the common elements of one set from another set.

---

Let's compare this with a sequence, which is an ordered set and may now have repetitive elements differentiated by where in the sequence they lie. This makes all of the previous operations --set union, set difference, and subsets-- no longer apply. We can replace them with similar ideas, but they will be different.

Def 5: A sequence, X, is a set which contains a list of ordered (now not necessarily unique) elements or "slots". In other words, it has a first slot, a second slot, and a so on. Given a sequence, X, of size N, we may define (X₁, X₂, X₃, ...) =/= (X₂, X₁, X₃, ...). In other words X₁ is in the first slot, X₂ is in the second slot, and so on.

But now addition of sequences is ill-defined. We can add each slot-wise, if each slot contains a number, anyways. Or we could add each slot up as a set (and thus X_i, for the i^th slot, would be a set). Or we could concatenate the sequences, so, X + Y = (X₁, ... , X_N, Y₁, ..., X_M), so sequence X of size N and sequence Y of size M. So you have to tell me how we're adding the sequences, explicitly.

---

If you're okay with the above, then let's proceed.

Well, I'm not really certain what name it would go by, but I think I can illustrate it.

-Fish example-
In the OP a meter as the basis for it's example. That is the length of a nice size fish. If I wish to give you a whole fish, and keep one for myself, the OP implies that all I need to do is chop it up into small enough pieces, then I can give you a whole fish, without decreasing the amount of fish I have. (ask if you need more explanation)

Let's come back to this later, hopefully after we have some agreement on the current issue.

I am under the assumption that math is the same every time you do it correctly, is that a safe assumption?

Sure, as long as you have clearly defined what it is that you're doing and that definition doesn't depend on the number of times you've done something.

(e.g. adding an odd number changes the evenness and oddness property depending on how many times you've added the odd number: 2, 5, 8, 11, etc, when you add 3 each time that changes. But yes, adding 2 + 3 will equal 5, every time.)

so, if you take 1 and you add another 1, then you will always get 2
Also, when you have 1 and you remove 1, then you will always get 0.
This process is irrelevant to who does it, or where it is done, or the order of the removal(in the case of multiple elements).

Okay, but we need to deconstruct "add" and "remove" here. What do you mean by "add" and "remove"? Do you mean add the number 1 to the number 1, or the set {1} to the set {1}? Or are we creating a new sequence by concatenating them?

Adding 1: Because as counting numbers, 1 + 1 = 2. If we take the union of them as sets, however, we get {1} + {1} = {1}. If we're concatenating them as sequences, we'd get (1) + (1) = (1,1). Which, if any, of these do you mean?

Removing 1: What do you mean by removing 1? IF you mean subtracting as numbers, then 2 -1 = 1. If you mean removing the set {1}, then {1} - {1} = {}, the empty set. If you mean as sequences, then you've got to be a lot more clear. You can subtract the first entry (1,1) - (1,0) = (0,1) or you could subtract the second entry, (1,1) - (0,1) = (1,0). Or you could mean that you just delete a slot. But which slot? If my sequence is (1,2), it matters which slot I delete.

This is of course flowing from the logical law of identity. (correct/agree?) If not feel free to offer a reading reference for the logical basis of mathematics.

Sure the law of identity holds in math, yes, but only after you tell me what you mean by your words. Above, you need to define "add" and "remove." There's no violation of the law of identity if you haven't told me what A and what B is, otherwise how can I tell you if "A is true but B is false" is a contradiction? It's only a contradiction if B is A. But if B is only similar to A, then there's no violation of a law of identity, no?

So, if you have Infinity, and you subtract 1, you will always get the same answer.
But this is not true when you subtract infinity. Suddenly an irrelevant element to mathematics, becomes relevant, Namely the location of the subtraction.

The location of the subtraction matters, but only if we're talking about sequences. Do you mean that you're defining slot-wise addition and subtraction? Or are you deleting a slot?

Or are you removing {1} as a set difference and adding {1} with set unions? If so, position can't and won't matter.

This is evident by the fact that if you have infinity and you subtract infinity you can get any and all possible sums, based on what the location is of that subtraction. (though all the while subtracting the same amount.. from the same amount, IE in the same way and in the same sense.)
This is evidence that infinity violates the law of identity. It is both itself and not itself at the same time and in the same sense.

You can't subtract infinity from infinity. Do you mean that you're subtracting one infinite sequence from another infinite sequence? Like, {1,2,3,4,5,6...} - {2,4,6} = {1,3,5,7,...}. Both sequences are infinite, but they aren't equivalent infinite sequences. They have different entries in each slot!

In order for me to know if there's a violation in the laws of logic, you need to tell me what kind of addition or subtraction you're doing.

---

Quick Side Note: In my example, I'm taking an infinite sequence of particles, plus a particle outside. So each slot is taken up by a physical Newtonian particle, defined by its mass, its current position, and its current velocity. Then I gave equations to tell you how all of these evolve in time. When I "added a particle", I meant that I moved each particle to the next slot (in this case, by evolving them to the next position). Then the particle outside evolved into the first slot after a specific period of time. This is a kind of concatenated addition, although it was physically performed. I simply added a new member to the front of my sequence, much like (0) + (1,2,3,4, ...) = (0,1,2,3,4, ...). The initial state was (1,2,3,4,...) where each number represents a unique particle, but we ended up with the sequence (0,1,2,3,4, ...), which is a different sequence (because particle 0 =/= particle 1), so there's no violation of the law of identity.

---

But I would have to say that the challenge for the OP, if it wants to counter WLC, is that it must find a way to effectively reach the audience such as WLC does.

The simplicity of the presentation really tells you nothing about the content of the presentation. I understand that it's ideal to be clear, but "1 =/= 2 because I say so" is a simple argument to understand, whereas discussing the law of identity at length is more abstract but nevertheless more meaningful and argumentatively valid.

With that said, I feel that Clive and I have been rather clear for a discussion on infinity.

Joking aside, let me summarize some observations about the way infinite are used here.. and you can let me know if the observations are correct.

1) Infinite introduce a directional bias in regards to outcome. (IE the direction of movement of the tenants dictates if the hotel is full, overflowing with guests, or has rooms to spare).
2) Infinite introduce a location bias in regards to outcome. (IE the location of the division determines if it's possible to accomplish. Such that if you tried to cut an infinite hotel in half.. you could not find the center.. but if you take every other room out you have accomplished the task)
3) Infinite tell me that If I desire to give you a fish, I need only cut it small enough to feed the world. That one fish = infinity fish. (IE destroying what it means to be 1)

1.) A-Fullness is not a physical property of infinite sets. It's up to you to defend why it should be.
2.) Infinite sets, infinite sequences, or infinite additions of numbers?
3.) We'll discuss this after we address the points above.

[MindTrap028]

Thanks GP, and Clive for your patience.. I'll be re-reading your responses a few times before I even attempt a response.

You can't subtract infinity from infinity

This jumped out at me, and I understand and appreciate your help in fleshing out the whole subtraction aspect you bring up throughout your post.
That said, I'm confused (probably through my own lossey goosey use of terminology of subtraction).. what do you mean you can't subtract infinity from infinity?

[GoldPhoenix]

Thanks GP, and Clive for your patience.. I'll be re-reading your responses a few times before I even attempt a response.



This jumped out at me, and I understand and appreciate your help in fleshing out the whole subtraction aspect you bring up throughout your post.
That said, I'm confused (probably through my own lossey goosey use of terminology of subtraction).. what do you mean you can't subtract infinity from infinity?

In the numerical sense. As sets, fine. As sequences (where you specify what you mean by subtraction, i.e. subtracting elements or deleting slots), fine. But as numbers, infinity isn't a number and so this doesn't make sense, generally.

[Squatch347]

Firstly, it's ridiculous that you're still confused by the technique of proof by contradiction and are continuing to confuse it with the logical fallacy of begging the question.

I understand your frustration here. If this was what I felt I was coming up against I would be, likewise, annoyed. But this frustration arises from misunderstanding the argument. I am not saying that any time you offer a hypothetical premise that it is begging the question, that would be, of course, silly. I think we can offer each other a bit more credit than that.

Rather, what I put forward above is that there are problems with the conclusion of your OP. That an actual infinite exists as part of an argument towards its coherency is an assumption, on that we both agree. I noted that there were further problems with your argument if we were to grant this assumption. IE the rest of my objections.

But what I also pointed out was that this assumption is problematic.

In order for us to say that your argument shows, as it purports to, that actual infinities can possibly exist within this universe we need to validate those assumptions are possible within this universe. Otherwise all we are doing is saying that it would be possible to imagine actual infinities which has nothing to do with Craig’s argument.

My point is that we need to have some kind of explanatory mechanism as to how that actual infinity might have arisen in this universe in order for us to grant it as a reasonable assumption within this universe, and by ignoring such a question you are begging the question on the defense of that assumption not on the entire argument.

Secondly, the closest you've come to addressing the point of the OP is when you keep on bringing up quantum mechanics (Or rather you keep on pontificating about the size of the particles, which was addressed in the OP, and I reminded you that this was synonymous with quantum issues) and how my example (which was explicitly stated to not be realistic, again in the OP) doesn't conform with quantum mechanics

This isn’t a rebuttal of the issues raised, it is just grand standing. In fact, it doesn’t seem as if anything in your response actually addresses my argument (aside from your belief about the fallacy above). You can claim I haven’t actually addressed your OP, but even the most casual review of the thread will reveal that that is intellectually dishonest, and blatantly so given the sheer number of questions and points that have yet to receive a response. And to be clear, all you need to do to show me wrong here is quote the text where you have already addressed it.

1a)Do particles in our universe occupy space?

1b) Is a point particle a useful idealization, like a frictionless plane, or an actual thing?

1c)In QM, do we treat particles as a point that occupies no physical space?

2) It still has not been directly addressed, and perhaps that is ok, but it would seem wise to clarify that your counter example isn’t about saying that the entire part of Craig’s defense is incorrect, only that one specific sub-section of this argument (adding a point to an infinite set) is incorrect.

3) Fair enough, but you don’t have an objection to the process being carried on successively do you? It wouldn’t cause your scenario any issues to do so, right?

4) Does the total mass of the interval change from t=0 to t=1? What is that total mass for the interval at both points?

5) Hence why I argued that the structure of your hypothetical is such as to avoid the issue by simply removing the rooms and making a hallway. Perhaps another way to look at it is to say that we could put a screen in every room so that we can fit an additional guest in there. And we can always fit another screen into that room to further sub-divide it and pack another guest in. Which works well and good if we have non-physical guests that don’t occupy space, not so much if those guests need somewhere to sleep.

Thirdly, I can understand thinking that the example given in the OP is contrived --it's definitely contrived. But its contrivance doesn't render it's point false, as has been defended by myself at least twice now and, again, ignored twice now by you.

Nor have I said it must be summarily rejected because it is contrived. My request was two, simple points. First, that your argument is missing the hidden premise in that your scenario is analogous to our actual universe (Remember that WLC’s argument is that actual infinities are impossible within our universe). IE you need to at least be able to defend that the scenario universe complies with the features of our universe in areas relevant to WLC’s argument.

The second is that there are certain features of your argument that don’t seem to comply with our universe, and my responses have to been to point out why that may be the case and ask you to defend or elaborate on your position.

Simply standing your ground and saying (with no supporting reference) that you’ve already covered it or that I haven’t addressed your argument is just poor form.

[GoldPhoenix]

I understand your frustration here. If this was what I felt I was coming up against I would be, likewise, annoyed. But this frustration arises from misunderstanding the argument. I am not saying that any time you offer a hypothetical premise that it is begging the question, that would be, of course, silly. I think we can offer each other a bit more credit than that.

I would love to give you more credit and blithely accept your re-envisioned argument, Squatch, except you said:

"Your defense was: “Suppose this hypothetical universe starts out with an infinite number of particles?” That isn’t a defense GP, it is an assumption. My point was that we can’t simply grant this assumption and move along with the hypothetical. To do so, is quite literally, begging the question. You are assuming an actual infinite number of particles exist to show that it is possible for an actual infinite number of particles to exist."

Emphasis mine; I think past-Squatch was pretty clear. Which only leads me to conclude that either my reading comprehension is incredibly poor, you phrased your argument in an incoherent manner, or that you were, in fact, genuinely confused over the difference between begging the question and proof by contradiction. I'm pretty sure it's not the former two.

In order for us to say that your argument shows, as it purports to, that actual infinities can possibly exist within this universe we need to validate those assumptions are possible within this universe. Otherwise all we are doing is saying that it would be possible to imagine actual infinities which has nothing to do with Craig’s argument.

As predicted in my previous post, here's your fourth time pretending that I haven't addressed this point, and here's my fifth time reminding you that I already answered this objection.

It was preemptively responded to partially at the end of the OP itself, a different more thorough argument was given at the end of post #5, and in addition to those an additional alternative argument was offered at the end of post #10. I've quoted it once or twice to you already and summarily neglected to read it each time, so you can look up yourself this time and see if you want to respond.

Oh, and just so we're clear:

---

Do understand that if you do not quote the relevant sections of the OP/post #5/post #10 and directly address them, I will not be responding to you any further on this thread. If in your next response you refuse to do so, please consider this my final reminder that you have not responded, and are unwilling to respond, to the arguments forwarded to you.

---

Moving on:

My point is that we need to have some kind of explanatory mechanism as to how that actual infinity might have arisen in this universe in order for us to grant it as a reasonable assumption within this universe, and by ignoring such a question you are begging the question on the defense of that assumption not on the entire argument.

You seem to be confused over the distinction between "I barely assert as fact that some kind of explanatory mechanism for actual infinities are necessary" and "I have a valid argument for why there needs to be an explanatory mechanism for how actual infinities come about."

You can claim I haven’t actually addressed your OP, but even the most casual review of the thread will reveal that that is intellectually dishonest, and blatantly so given the sheer number of questions and points that have yet to receive a response. And to be clear, all you need to do to show me wrong here is quote the text where you have already addressed it.

And upon said casual reading of the thread, one can verify that I have not made this claim, and in fact my actual claim was that you have been systematically ignoring huge sections of my posts that are, in fact, my rebuttal to your arguments, all whilst you sit there and claim that I've never dealt with your arguments in any fashion. Yes, I would be surprised if the word "intellectually dishonest" didn't spring into the interested reader's mind.

1a)Do particles in our universe occupy space?

1b) Is a point particle a useful idealization, like a frictionless plane, or an actual thing?

1c)In QM, do we treat particles as a point that occupies no physical space?

Addressed in the OP, the end of post #5, and the end of post #10.

2) It still has not been directly addressed, and perhaps that is ok, but it would seem wise to clarify that your counter example isn’t about saying that the entire part of Craig’s defense is incorrect, only that one specific sub-section of this argument (adding a point to an infinite set) is incorrect.

As has already been stated, the second part of WLC's argument (or at least your interpretation of it) does not get triggered by anything in the OP. In other words, there's no successive additions --despite your bloviation to the contrary and refusal to defend them-- contained in my OP. Therefore, even supposing that WLC's second argument was considerably more powerful than the former (sadly it isn't), it wouldn't matter because my argument didn't violate anything stated in WLC's second argument. There's a reason why I didn't have any infinite successive additions over time, and it's because, despite your bare assertions to the contrary, I am in fact aware of WLC's positions on the subject.

3) Fair enough, but you don’t have an objection to the process being carried on successively do you? It wouldn’t cause your scenario any issues to do so, right?

The context for this statement seems to be missing.

4) Does the total mass of the interval change from t=0 to t=1? What is that total mass for the interval at both points?

This has already been answered twice, and variations of the question ("What's the momentum?" and "What's the energy?") were directly addressed in the OP, but I'll repeat myself one last time:

After applying the mass regularization scheme proposed in post #10, the mass inside of the interval is 2m at the beginning, and 3m at the end of the time interval. The total mass in the universe at the beginning is 3m, coming from the 1m for the outside particle and 2m for the particles inside the interval. The total mass at the end of the universe is just the 3m lying inside the interval. Thus there is no violation of the conservation of mass, nor as the OP states a violation in the conservation of energy, or momentum.

5) Hence why I argued that the structure of your hypothetical is such as to avoid the issue by simply removing the rooms and making a hallway. Perhaps another way to look at it is to say that we could put a screen in every room so that we can fit an additional guest in there. And we can always fit another screen into that room to further sub-divide it and pack another guest in. Which works well and good if we have non-physical guests that don’t occupy space, not so much if those guests need somewhere to sleep.

This is just rephrasing of the "But don't particles have finite size?" shtick, which has been addressed.

Nor have I said it must be summarily rejected because it is contrived.

Nor was it forwarded that you had.

Simply standing your ground and saying (with no supporting reference) that you’ve already covered it or that I haven’t addressed your argument is just poor form.

I agree that it would be odiously poor form for a debater to do this, yes. I'm pretty happy that neither Clive nor MT has done this.

[MindTrap028]

Are you trying to say there's a conservation of mass issue?

No, more like when you give half a fish away.. it doesn't matter the size of the pieces you are going to be left with half a fish left.
More of an observation of reality, that I can repeat as much as you like (though you may have to provide some of the fish).

Though, in regards to the mass issue. I would think that if you could get an infinite number of fish from a single one fish, then the question is begged..Why aren't fish black holes?
What about slicing them increases the total mass?

I take it "the location of the amount removed" just means "which particular objects you choose to remove".

Well, we have a bit of confusion here. When you say which particular objects, It makes me think of removing Bob from the guest list no matter where he may happen to be.
Where as when I think of the location of the amount, it is more like the location regardless of which guest resides there.

In the case of a hotel, when removing an infinite amount of guests, the end total is based on the distance from the front door and where you start.

Sometimes properties/computations are affected by location, sometimes they aren't. For instance, in probability and statistics, if you're using the uniform distribution then your probabilities depend only on the length of the interval, not its position; no other probability distribution has this feature.

Sure, but here we are talking about subtraction, yes? The idea that location of the subtraction in a set is relevant to the end total, well that seems to be a deviation from the norm right?

What contradiction, if any, are you trying to identify? It's true that arithmetic on the cardinals is not isomorphic/identical to arithmetic on the real numbers. But there's plenty of algebraic objects whose operations aren't isomorphic/identical to the reals.

o.k.. .. I'm going to try and repeat this to you so I understand it.
Now cardinal numbers have to do with things like room numbers. You can have only 1 numeric room, with the cardinal designation of 1000.

I'll assume I am right here, but my answer is withdrawn if not.

Generally speaking, yes, but it depends on what you mean by "math". If you're taking samples from a random distribution, you might get different values each time, for example.

Sure.. but we are dealing with subtraction.
The objection being raised based on the inconsistency of answers given the same operation, based on elements that are typically irrelevant to that operation.

In the real numbers, order doesn't matter, since multiplication and addition are both associative and commutative. There are many algebraic structures that are either not associative, not commutative, or neither. In general, order does matter; associative and commutative structures are particularly well-behaved.

So the answer is that it depends on the algebraic structure you're working in.

I thought we were working with a fairly simple equation and thus algebraic structure.
Namely.. subtraction.

No, x=x doesn't imply x+y = y+x

1 = 1
2 = 2

1+2 = 2+1

That is incorrect? Because.. I hear you saying that isn't correct.

If you're doing subtraction in the reals, x-(y-z) =/= (x-y)-z, so it matters in what order you do your subtractions--i.e., do you subtract z from y, then subtract that result from x, or do you subtract y from x, and then subtract z from that result? But if you're doing addition, x+(y+z) = (x+y)+z, so order doesn't matter--i.e., you can add y to z, then add that to x, or you can add y to x, then add that to z.

You seem to be adding an element here.. what is Z in relation to our examples or even the example in the OP?

Matrix multiplication is associative, so A(BC) = (AB)C, but not commutative, so AB does not always equal BA.

For an operation that is commutative but not associative, consider the set S={Rock, Paper, Scissors}, and define multiplication as xy = the winner of x and y. So (Rock)(Paper) = Paper, (Rock)(Scissors) = Rock, etc. This operation is not associative, since [(Rock)(Paper)](Scissors) = (Paper)(Scissors) = Scissors, but (Rock)[(Paper)(Scissors)] = (Rock)(Scissors) = Rock. It is, however, commutative, since the winner of x and y is the winner of y and x.

Clive. you got to help me out buddy.
Is what you are describing anything like the Hotel? Because I don't think so.
I'm not certain if you are over-complicating things.. or if I'm just not getting how your response connects to the hotel objections.

Sure, if you're going with cardinals so your operations are defined in terms of set operations.

What are we doing with the hotel?

Well, you're using "location" in two different senses, here. One is algebraic, i.e. the location of "-" or the order of operations. The other is set-theoretic, e.g. which infinite subset of {0,1,...} you choose to remove. Both are relevant to the computations involved. So I don't know why you think that "location" is irrelevant to mathematics.

no, I wasn't quite referring to the location of subtraction in that sense.
I was referring to where in the hotel an infinit number of people are removed from the hotel, determines how many people remain in the hotel.
In one instance the remainder is infinite, in another only 5.
This is not an order of operations objection, as there is only one operation occurring.

Well, not in the same way, actually. You can subtract infinity from infinity in lots of different ways, if you're talking about taking relative complements of infinite subsets. You could take the evens out of the naturals, or the primes out of the naturals, etc.

The operations here are defined in terms of sets and cardinalities, which means certain formal expressions (like aleph0 - aleph0) are left undefined. This is not a contradiction, and follows (as you know) from the definition of arithmetic on infinite cardinals.

I don't know what an undefined actuality means.

This is manifestly false. Each cardinal is equal to itself.

Is infinity equal to itself?
Because it is manifestly and evidently not evidenced by empty rooms to a supposed non-decrease.

Well, this is why they leave subtraction undefined except in certain specific expressions. Transfinite arithmetic is defined in terms of set cardinalities, and (as you've seen) |A-B| doesn't just depend on |A| and |B|. So they leave |A|-|B| undefined.

It appears to me that, that is why there is a disconnect between the idea of infinits and reality.
Reality is defined, it is actualized, it is what it is (as they say in football).
To the extent that infinites are not defined, is the exact extent to which they can not be actualized.
If one operation that can occur in the real world is not defined.. then it can't be considered to be actualize-able. (I would think).

Well, there's lots of different operations you could define on the class of cardinals, or the extended reals, or the hyperreals, or whatever you like. You could call any of them "subtraction", if you wish.

But regardless of what operations you want to define, the question at hand is how infinite sets behave. And it seems clear that you have the following:

|{0,1,2,...} - {1,2,...}| = |{0}| = 1
|{0,1,2,...} - {5,6,7,...}| = |{0,1,2,3,4}| = 5
|{0,1,2,...} -{0,2,4,6,...}| = |{1,3,5,7,...}| = |{0,1,2,...}|

So it seems clear that |A-B| doesn't just depend on |A| and |B|.

The arithmetic here is defined in a way that depends on the behavior of the sets involved, i.e. in the cardinals, 2+3=5 because whenever two disjoint sets X and Y are such that |X| = 2 and |Y| = 3, we have |X U Y| = 5. aleph0 - 1 = alpeh0 because whenever |X| = aleph0 and |Y| = 1, |X-Y| = aleph0. aleph1-aleph0 = aleph1, since whenever |X|=aleph1 and |Y| = aleph0, |X-Y| = aleph1.

The problem with expressions like |X| - |X| when your operations are defined in terms of the behavior of the sets involved is that |X-Y| can be any cardinality less than or equal to |X|, even when |Y| = |X|.

But you could come up with different operations if you like. You don't have to base |X|+|Y| on the behavior of |XUY|, or |X|-|Y| on the behavior of |X-Y|.

I am not sure how that addresses my question.

Well, if I knew that you were using "-", "5", "4", "=", and "10" with their standard definition, I would give you a proof that 5-4 = 1, and 1 =/= 10.

Perhaps you could, but that would just be intuitively true, on pluto... it possibly doesn't work like that. (or so a defender of such a concept may say)

But there's lots of special exceptions in mathematics. x/y is defined--except for y=0, whoops. Every prime is odd--except for 2, whoops. Every integer doesn't have a multiplicative inverse in Z--except for 1 and -1, whoops.

Here we are talking about subtraction. Are there special exceptions to subtraction based on location (IE not order of operations location.. but location of the event).

Here the best understanding I can think to offer is to say that if you work a problem from right to left, you get one answer.
But if you write the problem backwards, AND work it backwards.. you get another answer.

Or, if you are holding apples, If I take it from your left hand you will have one remainder, and if I take it from the other, you will have a different remainder.

The answer is it depends on the context.

Thanks for this portion.

This is because there is a one-to-one function between any two countably infinite sets, and the set of all rooms whose number is greater than n is countably infinite.

There is an idea, and there is a reality.
If there is an empty room, that is the reality that there is not a 1 to 1.
If there is no room empty, then that is the reality that there is a 1 to 1.



So I consider the idea of it being both sufficient and insufficient in the same sense and at the same time evidenced and supported.
Is that fair?

I don't understand what you're trying to identify as problematic, here.

If you have a countably infinite list, you can leave off any finite portion of the list and still have countably infinite members on the list, which you can then assign to rooms using a one-to-one function since there are countably infinite rooms.

"Moving in one direction" means assigning the first guest starting at the nth room, and "moving in the other direction" means assigning the first room starting at the nth guest. As you would expect, "moving in one direction" leaves off rooms 1 through n-1, and "moving in the other direction" means leaving off guests 1 through (n-1).

I'm not sure how this means that "the direction of the math is relevant to the end sum" in any kind of problematic fashion. If you leave off rooms, you have extra rooms. If you leave off guests, you have extra guests.

My objection is the sufficiency argument, also the idea that location can be a relevant factor in reality.
Basically, infinity is not an amount capable of being actualized as it becomes a meaningless term.

Not every algebraic object has associative and commutative operations. Location is relevant to determining the probability that a random variable will fall into an interval of a certain length, for example.

Unless that is the operation being discussed, I'm not certain of it's relevance.

--------------------------------------

--------------------------------------

That may look like a bit of text, but it's just three points, really: A set is a collection of unique elements with a given size. We may make a new set by adding the unique elements of two previous sets. We may also subtract one set from another by removing the common elements of one set from another set.

O.k. so there is a lot going on in the thread, and honestly you and clive are repeating or mirroring each others points some.

But here I have a slight objection, and one I think that is different than WLC, sooo... with that in mind.
When you say a set is a list of objects, that is all well and good when we are talking about litteral objects.
but here we are talking about numbers, and numbers are concepts that exist in the mind.
I may have a problem with saying a number exists that has not been conceived in the mind. And (...) doesn't exactly strike me as being worthy of calling it "conceived".

This objection is made complicated by the fact that there could possibly exist a number that can not be physically expressed in the known universe by writing very small on paper.
(which my point asked.. in what way can it be said to exist at all, much less in the set).

But now addition of sequences is ill-defined. We can add each slot-wise, if each slot contains a number, anyways. Or we could add each slot up as a set (and thus Xi, for the ith slot, would be a set). Or we could concatenate the sequences, so, X + Y = (X1, ... , XN, Y1, ..., XM), so sequence X of size N and sequence Y of size M. So you have to tell me how we're adding the sequences, explicitly.

Well, I'm totally willing to let you define the way things are being added and subtracted in your OP. But, if the slots are the rooms of the hotel, then I don't think we are destroying rooms.

Let's come back to this later, hopefully after we have some agreement on the current issue.

No problem

Okay, but we need to deconstruct "add" and "remove" here. What do you mean by "add" and "remove"? Do you mean add the number 1 to the number 1, or the set {1} to the set {1}? Or are we creating a new sequence by concatenating them?

Well, my addition subtraction issue is more of the raw mathmatical version as opposed to the set version.
But, I have objections to both.

The First is that of subtracting infinity from infinity, now you have said it isn't really a number, and that confuses me because it is being used to tell the number of rooms in the hotel, and the number of guests. (I know reverting to the hotel).
Here we can get any answer, or as you said it is not done mathmatically for whatever reasons.

The second is that of subtracting a given number of guests, which is a reflection of the above. If you subtract an infinite number of guests, the remainder is determine by the location of that subtraction, and again.. you can get any answer.
to which I am objection to the relevance of location to mathmatics.

Sure the law of identity holds in math, yes, but only after you tell me what you mean by your words. Above, you need to define "add" and "remove." There's no violation of the law of identity if you haven't told me what A and what B is, otherwise how can I tell you if "A is true but B is false" is a contradiction? It's only a contradiction if B is A. But if B is only similar to A, then there's no violation of a law of identity, no?

I know this question is a running theme, so I'll leave it to you to condense your response, as I want to be sure to answer the question as best I can.
First, I am open to you defining what is going on in your op, this is me more defering to you rather than attempting to shift the burden.
Second, as for myself, i am referring to the two different kinds of subtraction. The first being the idea version represented in mathematics, the other is the actualized version
which I contend your scenario demands that direction and location are relevant factors. Which, I am hesitant to accept as relevant factors.

The location of the subtraction matters, but only if we're talking about sequences. Do you mean that you're defining slot-wise addition and subtraction? Or are you deleting a slot?

Or are you removing {1} as a set difference and adding {1} with set unions? If so, position can't and won't matter.

So, I'll repeat (or rather just point to) my objection of all conceivable numbers appropriately being called a set to start with.
I don't think we are doing slot subtraction, as we are not removing rooms.. but now my response is long, and I may be confusing that.
So, subtraction is numerically from the total, and locationally on the list.

You can't subtract infinity from infinity. Do you mean that you're subtracting one infinite sequence from another infinite sequence? Like, {1,2,3,4,5,6...} - {2,4,6} = {1,3,5,7,...}. Both sequences are infinite, but they aren't equivalent infinite sequences. They have different entries in each slot!

In order for me to know if there's a violation in the laws of logic, you need to tell me what kind of addition or subtraction you're doing.

Well, I have to object that you can't subtract infinity from infinity. If all the people leave the hotel, and the hotel held an infinity number of people.. then you have just subtracted infinity.
That is how reality works, people are allowed to check out.

It strikes me that infinity is being used to describe an unending process rather than an existing whole.
Like the real numbers. Sure we can always add 1 more to whatever number, but that is a process, to say it can be actualized into reality as a whole is simply untrue.

Quick Side Note: In my example, I'm taking an infinite sequence of particles, plus a particle outside. So each slot is taken up by a physical Newtonian particle, defined by its mass, its current position, and its current velocity. Then I gave equations to tell you how all of these evolve in time. When I "added a particle", I meant that I moved each particle to the next slot (in this case, by evolving them to the next position). Then the particle outside evolved into the first slot after a specific period of time. This is a kind of concatenated addition, although it was physically performed. I simply added a new member to the front of my sequence, much like (0) + (1,2,3,4, ...) = (0,1,2,3,4, ...). The initial state was (1,2,3,4,...) where each number represents a unique particle, but we ended up with the sequence (0,1,2,3,4, ...), which is a different sequence (because particle 0 =/= particle 1), so there's no violation of the law of identity.

Actually, I think that is the part I understand the best

The simplicity of the presentation really tells you nothing about the content of the presentation. I understand that it's ideal to be clear, but "1 =/= 2 because I say so" is a simple argument to understand, whereas discussing the law of identity at length is more abstract but nevertheless more meaningful and argumentatively valid.

With that said, I feel that Clive and I have been rather clear for a discussion on infinity.

I dig that, and you guys have been helpful of course. I was simply making a note of overall effectiveness, not really critiquing. You really have no obligation to make an argument simple at all.

1.) A-Fullness is not a physical property of infinite sets. It's up to you to defend why it should be.
2.) Infinite sets, infinite sequences, or infinite additions of numbers?
3.) We'll discuss this after we address the points above.

for reference sake..

Your argument here is: "The statement 'This set is full' means that you can't add more members without changing the size of the set." Let's call that definition/notion of being full as being "A-full." The answer to your question regarding the nature of the A-fullness of infinite sets is: Not applicable. You're committing a category error; A-fullness does not apply to infinite sets. Infinite sets fail to obey that property, but no one here is disputing that. The question is what is actually broken by a failure of A-fullness to hold for a set of physical objects. You need to either:

#1 - A-fullness.
My objection here remains, that fullness applies to real hotels, and to the extent that it doesn't apply to infinity is due to a logical contradiction in the idea of infinity. Which logically negates it from being realized into the actual world.
#2 .. I think in the case of division.. all of them. In the case of subtraction the location of the guests, and in the case of subtraction it's non-application to the numeric infinity (When numbers and mathematics are supposed to reflect reality or possible worlds.)
-- I am further confused by the idea that infinity is not a number, yet mathematics is used as a proof.
#3 looking forward to that exchange.

[CliveStaples]

No, more like when you give half a fish away.. it doesn't matter the size of the pieces you are going to be left with half a fish left.
More of an observation of reality, that I can repeat as much as you like (though you may have to provide some of the fish).

The fish doesn't have infinite mass. Giving away half the mass of the fish might entail giving away, say, the biggest 100 or so particles. But you'd only have half the mass of the fish left.

Though, in regards to the mass issue. I would think that if you could get an infinite number of fish from a single one fish, then the question is begged..Why aren't fish black holes?
What about slicing them increases the total mass?

Nothing increases the total mass. Mass is conserved.

Well, we have a bit of confusion here. When you say which particular objects, It makes me think of removing Bob from the guest list no matter where he may happen to be.
Where as when I think of the location of the amount, it is more like the location regardless of which guest resides there.

In the case of a hotel, when removing an infinite amount of guests, the end total is based on the distance from the front door and where you start.

The end is based on what you removed. If you removed every guest in rooms 6 and above, you're left with guests in room 5 and below. If you remove every guest in odd-numbered rooms, you're left with guests in even-numbered rooms.

I'm still not clear on what your objection is here, or what contradiction you're identifying. I'm not sure what your argument is.

Sure, but here we are talking about subtraction, yes? The idea that location of the subtraction in a set is relevant to the end total, well that seems to be a deviation from the norm right?

(I assume by "end total" you mean "the cardinality of the relative complement".)

It's a deviation from how cardinality of relative complements works for finite sets. But it's precisely what we'd expect for infinite sets, in accordance with the proofs with which you are already familiar.

Your task here is not to show that infinite sets behave differently from finite sets in this regard. Rather, your task is to show that the behavior of infinite sets entails a contradiction. Explicitly, this means that you need to show that there is some proposition P such that the behavior of infinite sets entails P and ~P. Showing that infinite sets are weird or pathological doesn't show that there's a contradiction. Fractals are weird and pathological (many are continuous everywhere but differentiable nowhere), but chaos theory seems to be doing just fine.

o.k.. .. I'm going to try and repeat this to you so I understand it.
Now cardinal numbers have to do with things like room numbers. You can have only 1 numeric room, with the cardinal designation of 1000.

I'll assume I am right here, but my answer is withdrawn if not.

Yes, but there are also transfinite cardinals--the aleph numbers, the beth numbers, etc.

Sure.. but we are dealing with subtraction.
The objection being raised based on the inconsistency of answers given the same operation, based on elements that are typically irrelevant to that operation.

The answers are not inconsistent, MT. The class of cardinals is well-defined. Given any two sets A and B, |A-B| and |B-A| are defined. You don't get different answers for |A-B| based on geographical location or anything silly like that.

I thought we were working with a fairly simple equation and thus algebraic structure.
Namely.. subtraction.

1 = 1
2 = 2

1+2 = 2+1

That is incorrect? Because.. I hear you saying that isn't correct.[/quote]

No, that is not correct. I never said that any of those equations were incorrect.

Addition (and its inverse, subtraction) is pretty familiar in the context of the natural numbers, and even for the real numbers (where you can compute things like pi^pi).

Addition for cardinals extends these familiar definitions. Let's get into the algebra here:

The definition of addition in the natural numbers is typically done recursively, something like this:

(1) n+0 = n
(2) For k >= 0: n+(k+1) = (n+k)+1 ["k+1" here denotes the successor of k, i.e. the image of k under the canonical successor function]

Natural number addition is well-defined, i.e. a+b = c+d whenever a=c and b=d.

Addition for cardinal numbers is defined differently:

Let A and B be sets with empty intersection. Then |A| + |B| = |A U B|.

Cardinal addition is well-defined, i.e. |A U B| = |C U D| whenever |A| = |C| and |B| = |D| (with A,B mutually disjoint and C,D mutually disjoint).

Cardinal addition extends natural number addition, i.e. if |A| = n and |B| = m (and A,B disjoint), then |AUB| = n+m.

You seem to be adding an element here.. what is Z in relation to our examples or even the example in the OP?

z is just another variable. I was showing that the order of subtraction matters, so that your objection is incoherent since we shouldn't expect order of subtraction not to matter.

Clive. you got to help me out buddy.
Is what you are describing anything like the Hotel? Because I don't think so.
I'm not certain if you are over-complicating things.. or if I'm just not getting how your response connects to the hotel objections.

I am showing that order of operations is not, in general, irrelevant. So if you have an expression like a $ b $ c, where our operation is $, it does generally make a difference what you compute first, i.e. (a $ b) $ c is in general different than a $ (b $ c). Operations for which those expressions are always equal are called associative operations.

Part of the issue here is that your "location" objection is just about completely vague, so I honestly have no idea what problem you're attempting to show exists with cardinalities and relative complements. Order of operations / computations is one kind of sense in which you might say that "location matters", so that's why I gave examples showing that your expectation is wrong with respect to associativity.

What are we doing with the hotel?

This has to do with what the hell you mean by an expression like infinity minus 1. You have to define what "infinity", "minus", and "1" mean before you can give a sensible answer to a question like "What is infinity minus 1?"

For example, the definition of "infinity minus 1" in the extended reals is different than the definition of "infinity minus 1" in the cardinals (cf. the above definitions of addition). For ordinal numbers, "infinity plus 1" and "1 plus infinity" give different results!

You must always be clear on what precisely the relevant definitions are for the expressions you're trying to evaluate.

no, I wasn't quite referring to the location of subtraction in that sense.
I was referring to where in the hotel an infinit number of people are removed from the hotel, determines how many people remain in the hotel.
In one instance the remainder is infinite, in another only 5.
This is not an order of operations objection, as there is only one operation occurring.

Okay. What's the contradiction?

I don't know what an undefined actuality means.

I don't know what you're talking about, here. I never used the phrase "undefined actuality".

Is infinity equal to itself?
Because it is manifestly and evidently not evidenced by empty rooms to a supposed non-decrease.

You need to go into more detail, here.

Yes, infinity is always equal to itself--in the sense that for every cardinal X, X = X. As I said, the definition of cardinal equality is as follows:

Let A and B be sets. Then |A| = |B| if and only if there exists a bijection between A and B (i.e., a one-to-one map from A to B or a one-to-one map from B to A).

To show that |X| = |X|, we need to show that there is a bijection from X to X. The identity map i:X -> X where i(x) = x for all x in X is a bijection. So |X| = |X|.

Now, you haven't offered an actual argument, you've simply said that "empty rooms to a supposed non-decrease" is evidence that infinity is not equal to itself. I take it you mean to argue for something like the following:

(M1) Let X be a set. Then for every non-empty subset Y of X, |X-Y| < |X|.

Now, the definition of < for cardinals is based on the existence of injections, i.e. |A| < |B| when there exists an injection from A to B, and |A| =/= |B| (or, equivalently, that there is no injection from B to A). To prove (M1), you would need to show that for every non-empty subset Y of X, there is an injection from X-Y to X and |X-Y| =/= |X| (i.e. there's no bijection between X-Y and X).

Sadly, this is doomed to failure. Let N = {0,1,2,...}, Y = {0}. Then Y is a non-empty subset of X. X-Y = {1,2,3,...}. The function f: X-Y -> X where f(n) = n is a bijection between X-Y and X. Therefore |X-Y| = |X|. So we can see that (M1) is false.

It appears to me that, that is why there is a disconnect between the idea of infinits and reality.
Reality is defined, it is actualized, it is what it is (as they say in football).
To the extent that infinites are not defined, is the exact extent to which they can not be actualized.

Infinities are defined. The class of ordinals and the class of cardinals are both well-defined.

If one operation that can occur in the real world is not defined.. then it can't be considered to be actualize-able. (I would think).

But the operations in question can be performed, namely for any infinite sets A and B the values |A-B| and |B-A| are well-defined. So you could always (theoretically) compute the size of the remainders after taking away a portion, i.e. if you remove everything from A that's in B, you're left with A-B, and |A-B| is well-defined.

What you couldn't do is something rather different than what you're suggesting. What you couldn't do is compute |A-B| if all you knew was |A| and |B|. If you wish to establish that this result is contradictory, then by all means provide a valid argument to that end.

I am not sure how that addresses my question.

I am telling you how the relevant operations on cardinals are defined so that you could (hopefully) understand that there's no reason we should think that every cardinal will have an additive inverse, and that the operations in question are defined in terms of functions between sets, which are well-defined--and thus won't give different answers depending on geography or anything silly like that.

Perhaps you could, but that would just be intuitively true, on pluto... it possibly doesn't work like that. (or so a defender of such a concept may say)

No, it would be formally derivable from the axioms that define the given operations and expressions. Whether or not the results are intuitive is irrelevant to whether the results are entailed by the axioms. Also irrelevant to the entailments in question is one's geographical position in the universe.

Here we are talking about subtraction. Are there special exceptions to subtraction based on location (IE not order of operations location.. but location of the event).

I don't know what you mean by "location of the event". It doesn't matter where you are when you try to figure out |A-B|. A and B matter.

Here the best understanding I can think to offer is to say that if you work a problem from right to left, you get one answer.
But if you write the problem backwards, AND work it backwards.. you get another answer.

I have no idea how this relates to cardinal arithmetic. The values |A-B| and |B-A| are well-defined for every pair of sets A,B.

Or, if you are holding apples, If I take it from your left hand you will have one remainder, and if I take it from the other, you will have a different remainder.

Are you trying to get at the fact that you can have |A| = |B|, but |N-A| =/= |N-B| when |A| and |N| are infinite? I still don't know what contradiction you're alleging, here.

There is an idea, and there is a reality.
If there is an empty room, that is the reality that there is not a 1 to 1.
If there is no room empty, then that is the reality that there is a 1 to 1.

I don't quite know what you mean, here. 1-to-1 functions exist between sets; they have a domain (input) and a co-domain (output). I'll try to figure out what you mean, but I'd really appreciate it if you'd start being more clear in your posts about what the hell it is you're talking about.

Perhaps you mean something like this:

Let G be a set of guests, and let H be a set of rooms. Let R:G -> H be a function--let's call it a room-assignment function, since it takes as input a guest in G and gives as output a room in H.

(M2) If there is a room in H that is not assigned to a guest in G, then R is not 1-to-1.
(M3) If every room in H is assigned a guest in G, then |G| = |H|.

(M2) is true. Since there's an element in H that R does not map to, R is not a surjection and hence cannot be a bijection. I don't know what contradiction you're attempting to establish, here. Are you suggesting that if R isn't a bijection, then |G| can't equal |H|?

(M3) is simply false. If G is uncountable and H is countable then |G| > |H| so there is no 1-to-1 map between G and H.

So I consider the idea of it being both sufficient and insufficient in the same sense and at the same time evidenced and supported.
Is that fair?

Not at all, no. My understanding is that your definition of sufficiency doesn't establish a contradiction--i.e. every set is either sufficient or insufficient, but never both.

For the sake of convenience, can you state your definition of sufficiency for sets?

My objection is the sufficiency argument, also the idea that location can be a relevant factor in reality.
Basically, infinity is not an amount capable of being actualized as it becomes a meaningless term.

Your location argument is incoherent at this point as a result of being overly vague. Your sufficiency argument is simply wrong.

Unless that is the operation being discussed, I'm not certain of it's relevance.

The relevance is as follows: it establishes that your expectations/intuitions with regard to the well-behavedness of operations are quite mistaken and should be revised.

[MindTrap028]

Thanks for the response Clive, but If my exchange with GP gets nearly as long.. I'm going to be lost.
I will try to take it in smaller bites, of course I run the risk of of leaving stuff out in the end...


----------A Fishy business---------

The fish doesn't have infinite mass. Giving away half the mass of the fish might entail giving away, say, the biggest 100 or so particles. But you'd only have half the mass of the fish left.

Well, the idea of "biggest particles" is not represented in the example of the OP.
The OP points to a real thing in the space of a meter.. I have simply applied it to fish.
Infinite fish is the logical conclusion.

You are subtracting half the fish.. but half of infinity = infinity.. so you have infinity so you still have a whole (no smaller) fish left.. you just re-assign the particles per the examples in the thread.

Nothing increases the total mass. Mass is conserved.

So you are expecting an infinite number of lighter fish?
Why? The parts were all equal and infinite. Infinity says we can subtract infinity (even half of it) infinity number of times without decreasing the total.

So either reality bars the example at all.. or you get infinite fish.


Lets call this the "Gravitational objection to infinity".
-----------------------------------------

[CliveStaples]

Well, the idea of "biggest particles" is not represented in the example of the OP.

The particles all have different masses, so some particles are "bigger" (in terms of mass) than others.

You are subtracting half the fish.. but half of infinity = infinity.. so you have infinity so you still have a whole (no smaller) fish left.. you just re-assign the particles per the examples in the thread.

You'll have just as many fish particles as you had before, but far less mass of fish.

So you are expecting an infinite number of lighter fish?
Why? The parts were all equal and infinite. Infinity says we can subtract infinity (even half of it) infinity number of times without decreasing the total.

The parts were not all equal, otherwise the fish would have infinite mass.

[MindTrap028]

The particles all have different masses, so some particles are "bigger" (in terms of mass) than others.

Such particles can not exist in reality due to point mass problems.
This was noted earlier in the thread.

http://scienceworld.wolfram.com/physics/PointMass.html

A geometric (0-dimensional) point that may be assigned a finite mass. Since a point has zero volume, the density of a point mass having a finite mass is infinite, so point masses do not exist in reality. However, it is often a useful simplification in real problems to consider bodies point masses, especially when the dimensions of the bodies are much less than the distances among them.

You'll have just as many fish particles as you had before, but far less mass of fish.
...
The parts were not all equal, otherwise the fish would have infinite mass.

This can only be true if point masses can exist or do exist. As far as I know they do not and can not, thus I think the point of "Gravitational objection to infinity" stands. That said... I'm no expert and I'm taking the word of a source when it was very difficult for me to find any information in relation to infinity at all. So, I welcome any education on this point why point masses can exist Or to why the source has it otherwise wrong.

----
Note* I'm under the impression that the OP doesn't ascribe mass to any of the points being discussed.

[GoldPhoenix]

MT: I can deal with your previous questions, although I think that Clive has done a good job at responding to you, and I think there's largely overlap so unless when I re-read it I think that there's something that needs specifically discussed further (or let me know if there's something specific you'd like addressed), I'll leave the previous discussions and focus in on this, because it's quite concrete what you're asking.


N.B. Squatch, you could stand to read and verify what's been presented here, as well, because it answers your questions about how the conservation of mass is specifically achieved in this model.

Well, the idea of "biggest particles" is not represented in the example of the OP.
The OP points to a real thing in the space of a meter.. I have simply applied it to fish.
Infinite fish is the logical conclusion.

You are subtracting half the fish.. but half of infinity = infinity.. so you have infinity so you still have a whole (no smaller) fish left.. you just re-assign the particles per the examples in the thread.


So you are expecting an infinite number of lighter fish?
Why? The parts were all equal and infinite. Infinity says we can subtract infinity (even half of it) infinity number of times without decreasing the total.

So either reality bars the example at all.. or you get infinite fish.


Lets call this the "Gravitational objection to infinity".

Okay, so your question appears to essentially be "But look, GP, what is you took away half of the particles, there's still an infinite amount of mass!" And then when you account for gravity, that will be a serious problem. There's several things to clear up here:


1.) Yes, you can remove an infinity ("half", as you're calling it) of particles away. But there's two mistakes you're making, which is ironically you aren't paying attention to the conservation of mass. There's a specific mass attached to each particle in my scenario (Again, in post#10, I specified that we could change the mass to give a finite result). In my modified scenario, the mass of the n^th particle is given by m_n = m 1/n^2, so the first particle has mass m, the next particle m/2, the next m/4, and so on. So the total mass inside of the interval of points between 1 and 2 is finite, and the total mass can be calculated by the geometric formula (see 2) to be 2m. Secondly, a pretty simple application of the squeeze theorem for series will tell you that if I remove particles --any particles-- the result will remain finite, and in fact I will exactly lower the total mass inside of my interval by the mass of the particle that I removed. But let's do it explicitly and clearly.


2.) For the record, the total mass at the beginning is 2m, as stated above. Again the relevant formulas can be found here, which I will make extensive use of here. Okay, so you're asking what happens when we remove half of the particles? That's a fair question, and it's a simple question to ask in my model. So let's disregard the particle on the outside for now (it won't play any role, it only shifts the mass by a constant +1m either way, but since we'll be leaving it alone it'll just contribute a +1m to both before and after). Then, you're question is "What happens if instead we were to move half of the particles outside of the interval?" To do this, we'll use the same velocity trick as last time, but now instead of moving every particle over, we'll cause the odd particles to say where they are, and shift all of the even particles over by 1. (It's left as a proof to the reader that if each interval measures a length L and we move each even particle over a distance 1 L in a time T, then the total momentum of the system is p = 3/4 m L/T; by the squeeze theorem, each term in the energy series is less than the terms in the momentum series, so while it's completely left up to the reader to show the precise energy, the total energy is necessarily finite by the squeeze theorem.) Thus they will sit on the new points, with one sequence sitting on Y_n on the interval [2,3] and the old stationary particles sitting on the interval [1,2]. Specifically, these are given by:

Particles moved to interval [2,3]:

( 2, 2+1/4, 2+1/4+1/16, ...)
Explicitly, y_N = 1 + Sum[(1/2)²ⁿ;0 to N] = 1 + (1 - (1/4)ⁿ)/( 1 - 1/4)

Particles staying on interval [1,2]:

( 1+1/2, 1+ 1/2+1/8, 1+1/2+1/8+1/32, ... )

Explicitly, x_N = -1 + 2*Sum[ (1/2)²ⁿ; 0 to N ] )) = -1 + 2 (1 - (1/4)ⁿ)/( 1 - 1/4).

It may be verified by the reader that the specific formulas are correct (Just plug in the values I gave you, and you'll see that each formula at the end reproduces the correct values). Now we can straightforwardly sum over the masses of the two different sequences of particles along the two respective intervals (each containing an infinite number of particles) by counting the masses, which again the reader can verify for themselves is correct by recalling the mass of the n^th particle is given by m_n = m (1/n)² and noting that all the even particles go on [2,3] starting with and all of the odd particles go on [1,2]:

m_1st = lim[n -> infinity] of "2m(1 - (1/4)ⁿ)/( 1 - 1/4) } -2m" = 2/3m

m_2nd = lim[n -> infinity] of " m(1 - (1/4)ⁿ)/( 1 - 1/4) " = 4/3 m

And the sum of each individual sequence is:

m_total = m_1st + m_2nd = 4/3 m + 2/3 m = 6/3 m = 2m

Which is the same total amount of mass, as promised from the outset of the problem. Thus mass, energy, and momentum are explicitly conserved, despite there being an infinite number of particles. And in fact, I have shown that I can move an infinite number of particles around, and all physical quantities remain finite and consistent with conservation laws.

---

N.B. For those interested in the sequences, the even sequence looks like:

1
1+ (1 + 1/4)
1 + (1 + 1/4 + 1/16)
1+ (1 + 1/4 + 1/16 + 1/64)
.
.
.
1 + ( 1 - (1/4)^n )/(1 - 1/4)

(Here we've used the trivial identity that 2²ⁿ = 4ⁿ). And the odd sequence looks like:

1 + 1/2
1 + 1/2 + 1/8
1 + 1/2 + 1/8 + 1/32
.
.
.

Which one can see is equivalent to, after deleting the first slot that's nothing but zero (Which won't affect series but will affect the sequence):

= -1 + 2( 1 + 1/4 + 1/16 + ) = 1 + 2(1/4 + 1/16 + 1/64 + ... ) = 1 + 1/2 + 1/8 + 1/32 + ...
= -1 + 2( 1 - (1/4)^n )/(1 - 1/4)

But we don't want to double count the 1 when we add masses on each of sequence (because there's a fake slot with a 0 in it), so we need to subtract off an m on the odd sequence since there's no particle at 1 (It got moved over to 2 and is in the other sequence), and thus there's an over all (sum)-2m instead of (sum)-1m when counting the masses for the first slot. Another way of saying this is that on the odd sequence, the first mass is m/2, not (1+1/2)m, so you can see the double counting here and the need for subtracting the extra one. It's just due to how we've juggled the indices on the summand to put everything into the form of the geometric formula.

---

To reiterate the point of the OP, the reason I chose this less-than-physical example (Newtonian mechanics withouth gravity) is because you can ask a lot of commonsense questions like MT is doing, and you can actually ask them and get calculable answers. So while it doesn't identically model Nature (We don't currently have a theory for doing that, and as I said to Squatch, we don't know if such a theory will or won't include point-particles), it does allow us to explore our intuitions in a simple, straightforward, unambiguous, and exactly calculable manner, and see how they hold up to simplified physical scenarios. At any rate, WLC's "inconsistency" really, I think, should have shown up by now.