Axiom of Extensionality and 'if and only if' statements

Question

I was looking at the Axiom of Extensionality:

$\forall A\,\forall B\,(\forall X\,(X\in A\iff X\in B)\Rightarrow A=B)$

I somewhat understand the 'if and only if' statement, but I am having (a lot) of trouble formalizing it. In my head, I imagine a situation like this:

If $X$ is the entire set of $A$ and $X$ is the entire set of $B$, $A=B$.

And the statement holds.

But what about this:

Say $A = [1,2,3]$ and $B = [1,2,3,4,5]$ and $X = 1$. $X$ is an element of $A$ and $B$, but $A$ is not $B$.

Clearly, my counterexample is wrong, and it has something to do with the 'If and only if' statement. But I do not understand where the logic is wrong.

I suggest you clean the slate and restart by: If, for all $X$, $X$ is an element of $A$ if and only if $X$ is an element of $B$, then $A=B$. — , Sep 26 '18 at 04:08
@SaucyO'Path I could rewrite the notation in words, but my problem is not that I don't know how to read the statement; it is that I can only understand it a certain way, and even then, my understanding does not hold. I am asking where my understanding is wrong, not how to write my statement. — Descartes Before the Horse, Sep 26 '18 at 04:12

score 4 · Accepted Answer · answered Sep 26 '18 at 04:16

I think you are confused about how this is parenthesized. It should be parenthesized as $$\forall A\,\forall B\,([\forall X\,(X\in A\iff X\in B)]\Rightarrow A=B)$$ and NOT as $$\forall A\,\forall B\,(\forall X\,[(X\in A\iff X\in B)\Rightarrow A=B]).$$ This means that to conclude that $A=B$, you need to know that $X\in A\iff X\in B$ for all $X$, not just for one single $X$.

score 0 · Answer 2 · answered Sep 26 '18 at 04:21

0

I found an older post that helps with this. Here it is, if anyone is ever interested.

answered Sep 26 '18 at 04:21

Descartes Before the Horse

7,386

kinsey_eng · Answer 3 · 2021-11-19T17:02:59.307

I believe the confusion here is around the definition of $X$.

$A$ and $B$ are presumed to be any arbitrary sets within the Domain.

$X$, on the other hand, is presumed to be something else. The only way I have been able to make sense of this definition is to assume something like:

"$X$ is a set containing the entire Domain of Discourse."

I think the presumption of the definition of $X$ to be such a "universal" set, or all possible sets in the domain (power set), is what enables this axiom to be generalized to the definition of equality.

(But without this interpretation of $X$, the Axiom seems to lose its meaning...so it appears to be a crucial presumption.)

I would welcome anyone to provide a more concrete presupposition of $X$.

Axiom of Extensionality and 'if and only if' statements

3 Answers3

Linked