# Axiom talk:Axiom of Extension

Can I add "The order **and multiplicity** of the elements in the sets is immaterial." ? I'm trying to think of a better word than multiplicity since that has certain connotations, but I think it's an important point. Or do you think it's sufficiently implied in what's on the page? --GFauxPas 11:32, 27 December 2011 (CST)

- No, because at the level of definition of axiomatic set theory the concept of multiple instances of a single element does not arise, it can't happen in this context. I think. --prime mover 14:52, 27 December 2011 (CST)

## Alternate Formulation

Is the alternate form a weak form of the standard one, or is it completely different? The statement $A \in x$ is bothering me; I wanted it to be $x \in A$, since we're quantifying over $x$. --GFauxPas 18:15, 11 August 2012 (UTC)

- The alternate form is for set theories that define $=$. You can change the variables from $A$ if you want. --Andrew Salmon 19:14, 11 August 2012 (UTC)
- It's not that, it's just that I'm not sure what the scope of the quantifier is. --GFauxPas 20:36, 11 August 2012 (UTC)

- In either case, the quantification is over sets. I get the impression that the first formulation takes $A, B$ to be sets (I'm quite sure about that, in fact), and the second admits $A,B$ to be classes. However I lack (published) sources for either, hence I can't be certain. --Lord_Farin 00:54, 12 August 2012 (UTC)

- The second admits $A,B$ only as sets. Otherwise, this would become an axiom schema. We
*can*prove an equivalent statement for classes using this axiom, however. --Andrew Salmon 02:46, 12 August 2012 (UTC)

- The second admits $A,B$ only as sets. Otherwise, this would become an axiom schema. We

A useful rule of thumb here is: if there exists ambiguity on a page which is resolved by the answering of questions on the talk page, then the original page needs to be updated to include the extra explanatory information as developed on that talk page. --prime mover 05:00, 12 August 2012 (UTC)

## set theories that define equals

Why is it not possible to use:

- $\forall x: \left({x \in A \iff x \in B}\right) \iff A = B$

in a set theory that defines equals? Surely this very sentence defines what $=$ means? --prime mover (talk) 06:19, 28 August 2012 (UTC)

- Because that would be the definition of equality and would no longer be an axiom. However, such a system is incapable of proving $A = B \land A \in C \implies B \in C$. --Andrew Salmon (talk) 02:09, 29 August 2012 (UTC)

- Can you explain? Like, write a page demonstrating how? --prime mover (talk) 05:25, 29 August 2012 (UTC)

## Primitive

PM, read the definition it links to. $=$ is a primitive symbol in every formal language containing it because it is just one character. So whether or not I changed the link to the right thing, I *certainly* changed it from a wrong thing. --Dfeuer (talk) 20:18, 11 March 2013 (UTC)

- Sorry: it's wrong why? The ZF axioms are at base a formal system. --prime mover (talk) 20:38, 11 March 2013 (UTC)

- Because $=$ is certainly part of the alphabet of any formal language using it. The distinction is between theories in which it is assigned a meaning and ones in which it is instead described axiomatically. That is, whether it is or is not an undefined term. --Dfeuer (talk) 20:55, 11 March 2013 (UTC)