EDIT: I corrected a confusing point raised by Clive below.

I don't care if you don't formally prove statements, but I don't believe that there's an honest way to have this conversation without you being more explicit than you are below. I don't care if you spell out your reasoning here with words or equations, but I do need the reasoning to be explicit and not implicit. In other words, I need you to try to spell out what the operations are that you're writing.Originally Posted byMT

1.) This is vague, and by this I mean that your definitions and reasoning are essentially entirely implicit rather than explicit. You indicate that X = 1 or 2, so I can only assume that you mean that X is a counting number (1,2,3,..., etc, the numbers that can be used to count the number of objects in finite sets). I amassumingthen that you mean that "infinity" in the expression "infinity - infinity" is meant to be something that is used to count the number of objects in infinite sets, too. In other words, "infinity" is "aleph-null", the name for the "size" of infinite sets. In which case, I am forced to assume that you mean "-" is supposed to be related to a kind of set difference. I am going to assume these things for now, please correct me if this isn't the case.Firstly, you appear to have an order of operations error.Let's take two infinite sets to illustrate this.

X = {1,2,3,4,5,6, ...}

Y = {2,4,6,8,10,12, ...}

So, let's denote the size of the sets to be given by Size(X) = the size of X. You seem to want to demand something like:

Proposition: "For all a sets U with a subset V, then Size(U - V) = Size(U) - Size(V)."

This isn't a valid theorem; it holds for finite sets, but it doesn't hold for infinite sets. Using the infinite sets defined above, you seem to be arguing something along the the lines of:

Size(X - Y) = Size({1,3,5,7,9,....}) = Size(X) = aleph-null

Now, the crucial failure in your reasoning is that you're assuming that we can do simple arithmetic (addition, subtraction, and so on) with aleph-null, and that contradiction in doing ordinary arithmetic results in us concluding that aleph-null is itself contradictory. But that is incorrect because no one is saying that you can do arithmetic with infinity (add, subtract, etc) in terms of just playing around with the sizes of the set.

So you want us to accept that:

Size(X - Y) = Size(X)

is a contradiction, because you're going to apply (the incorrect) formula on the left-hand side of the equation to get:

Size(X) - Size(Y) = Size(X)

Subtract a Size(X) from both sides to yield:

==> Size(Y) = 0

AND have us assert

Size(Y) = Size({2,4,6,8,....}) = aleph-null =/= 0

And THEN, yes, we would derive a contradiction.

The problem is straightforward:

1.)Size(X - Y) =/= Size(X) - Size(Y)in many instances, including this one. Crucially because:

2.)Aleph-null does not have arithmetic properties; the expression "Size(X) - Size(Y)" is undefined. In other words, you aren't allowed to subtract aleph-null (the size of X) from both sides to get the equation Size(Y) = 0. Aleph-null isn't a counting number, there isn't a way to subtract it, and therefore the proof is erroneous.

2.)Secondly, the valid part of your reasoning seems to stem from the intuition described in the following scenario. The part that may disturb you if you try to do what you're doing and boil everything down to just the size of an infinite set (again, you'll run into similar conceptual problems even for finite sets) is that you think that:

Size(A - B) = 0 <==> A = B.

This is almost true. We need a small clarification, and then it's absolutely true.

In other words, the correct formulas that your intuition seems to be trying to nail down are:

Def: Let "B is the subset of A" mean that "There are no objects inside of B that aren't also in A." (Note: This doesn't exclude B=A, it just checks that there's nothing in B that isn't also in A.)

Theorem: "If B is a subset of A and Size(A-B) = 0, then A = B."

Theorem: "If B is a subset of A, and Size(A-B) =/= 0, then A =/= B."

In set theory, you can prove that this statement is true. And guess what?Infinite sets obey this property.

But what is not true is that:

Non-Theorem: "If B is a subset of A, and Size(A) = Size(B), then B = A."

If A is finite, then B is finite, and then Size(A) is a natural number, in which case operations like "plus" and "minus" are defined. Then,and only then, the proof give in part (1) becomes valid because "Size(A) - Size(B)" is now defined because Size(A) is a natural number and Size(B) is a natural number, and we can define subtraction between the sizes.But if either A or B were infinite, then the formula "Size(A-B) = Size(A) - Size(B)" not only isn't true, it doesn't mean anything because it's not defined.

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