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 (X_{1}, X_{2}, X_{3}, ...) =/= (X_{2}, X_{1}, X_{3}, ...). In other words X_{1} is in the first slot, X_{2} 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 = (X_{1}, ... , X_{N}, Y_{1}, ..., 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.

Originally Posted by

**MindTrap028**
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.

Originally Posted by

**MindTrap**
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.)

Originally Posted by

**MT**
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.

Originally Posted by

**MT**
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?

Originally Posted by

**MT**
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.

Originally Posted by

**MT**
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.

Originally Posted by

**MT**
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.

Originally Posted by

**MT**
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.

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