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So, I've been reading through Charles Pinter's A Book of Set Theory and I've become rather confused about a definition he has made:

Let $A$ and $B$ be classes. Then, we define:

$$A = B \iff (\forall X)[(A \in X \implies B \in X) \land (B \in X \implies A \in X)]$$

What I'm really confused by is the fact that he, then, goes on to state the Axiom of Extent as the defining feature of class equality. So, he states it as follows:

Let $A$ and $B$ be classes. Then:

$$A=B \iff [(x \in A \implies x \in B) \land (x \in B \implies x \in A)]$$

That seems rather strange to me. It seems like there are two different definitions of equality here that don't quite match up in my mind.

Could someone give me some intuition on this? Like, why does it work? If it doesn't, then is it just wrong and, if so, why?

Mousedorff
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  • What is the Axiom of Extent in Pinter's book? Is it valid for sets or classes? – Henno Brandsma Mar 21 '20 at 14:53
  • See the discussion in Axiom of Extensionality – Mauro ALLEGRANZA Mar 21 '20 at 14:55
  • See also the post Equality in set theory – Mauro ALLEGRANZA Mar 21 '20 at 14:56
  • It mentioned, in an earlier paragraph, that it would go through the differences between a set and a proper class later. It has stated the axiom of extent for classes, it seems. @HennoBrandsma – Mousedorff Mar 21 '20 at 14:59
  • @MauroALLEGRANZA I've read through the wikipedia link, I think it'll take a bit more time to parse through the answers to the other question. From my understanding, it seems like the first statement that I have above is the definition of equality (so it's defining the equals sign in set theory) while the Axiom of Extent specifically defines equality of sets.

    My question still remains: why is this necessary? Why not just use the Axiom of Extent?

     – Mousedorff Mar 21 '20 at 15:05
  • As said in Wiki and in the post linked there are two basic approaches: (i) underlying logic is FOL with equality; this means that we have already the axioms for equality and set theory add the Extensionality Axiom, asserting that if two sets have the same elements, then they are equal. – Mauro ALLEGRANZA Mar 21 '20 at 17:03
  • (ii) underlying logic is "pure" FOL (no equality). In this case we have an Axiom defining equality (that you have in your post) that "incorporate" Extensionality. From it the usual properties of equality (refl,symm,trans) are proved. – Mauro ALLEGRANZA Mar 21 '20 at 17:05
  • Ah, okay, I see what you mean. I'll have to mull over this and maybe ask more questions if it does come up again. – Mousedorff Mar 24 '20 at 08:35

1 Answers1

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Here is a rather incomplete answer.

We want equality to have the property of "substitutivity": For every statement $\beta(x),$ if $\beta(x)$ holds and $x=x'$ then $\beta(x')$ also holds. This property will be guaranteed by the $(\Rightarrow)$ direction of your first display, once we assume $=$ is reflexive, symmetric and transitive.

Source: Section II.2 of Fraenkel, Bar-Hillel & Levy, Foundations of Set Theory, 1973, 2nd rev ed, Elsevier. (See especially page 22.)

(Also, the following related post may or may not be helpful: axioms of equality.)

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