How can we verify if a structure statisfies ZFC or not?

Question

I'm studying set theory which leads me to some confusions of model theory.

For the context, I know that the set $\mathbb{N}$ with the usual operations $(0,+,×)$ is a model of Peano arithmetic. It seems to me it is quite obvious (except for the axiom of mathematical induction, which I don't know how to prove) to verify that this structure satisfies the axioms of Peano. For example the 2 following axioms:

1. $0$ $\in$ $\mathbb{N}$ (I point at the number $0$ in the structure)

2. if $n \in \mathbb{N}$ then $n+1 \in \mathbb{N}$ (I point at 2 numbers $n$ and $n+1$ in the set $\mathbb{N}$ of the structure)

Concerning ZFC, I know that it has many models. My concern is how can we verify that a given model satisfies ZFC ? It seems to me that we need to verify if the given model satisfies all the axioms of ZFC or not (then it will automatically satisfies all the sentences of the theory). But I can't think of a way to actually verify if a structure (i.e. a model) satisfies or not an axiom of ZFC, for example :

Axiom of extension: Two sets are equal if and only if they have the same elements.

Axiom of specification: To every set A and to every condition $S(x)$ there corresponds a set $B$ whose elements are exactly those elements $x$ of $A$ for which $S(x)$ holds.

As for me, these axioms are so obvious, much more obvious than those of Peano. How can we prove that a given structure satisfies these obvious axioms ? Or in general, how can we prove that a given structure satisfies ZFC ?

Could you please enlighten me ? Thank you very much for your help!

Answer 1

I'll go through the first one, and show that ZF proves that any Transitive Proper Class is a Model of The Axiom of Extentionality.

In fact, a much weaker theory T = {Axiom of Extentionality} proves that any transitive Proper Class is a Model of the Axiom of Extentionality.

Recall that a Class A is transitive iff for any a in A, a is a subset of A.

The other Axioms follow a similarly styled argument.

Definition:

Let M be a class.

M is a model of a set of sentences S, iff each sentence in S is true in M.

Definition: A sentence $\phi$ is true in a Proper Class M iff $\phi$ is true relativized to M, which we refer to as $\phi^M$

We define $\phi^M$ by cases depending on the complexity of $\phi$

Case 1: $\phi$ is Atomic, so $\phi$ is x = y or x$\in$y

$\phi^M$ = $\phi$

Case 2: $\phi$ = $\exists$x$\varphi$

$\phi^M$ = $\exists$x$\in$M$\varphi^M$

Case 3: $\phi$ = $\phi_1$ $\wedge$ $\phi_2$

$\phi^M$ = $\phi_1^M$ $\wedge$ $\phi_2^M$

Case 4: $\phi$ = $\neg$$\varphi$

$\phi^M$ = $\neg$($\varphi^M$)

Theorem ( Under a Theory T = {Axiom of Extentionality})

Let M be a transitive Proper Class and S = {Axiom of Extentionality}, then M is a Model of S.

Proof:

Recall that the Axiom of Extentionality is $\forall$x $\forall$y[ $\forall$z(z $\in$ x $\iff$ z $\in$ y) $\rightarrow$ x = y]

We prove that the axiom of Extentionality is true in M

Consider the Axiom of Extentionality relativized to M,

$\forall$x$\in$M$\forall$y$\in$M [ $\forall$z$\in$M (z $\in$ x $\iff$ z $\in$ y) $\rightarrow$ x = y]

Specialize x$\in$M and y$\in$M

$\forall$z$\in$M (z $\in$ x $\iff$ z $\in$ y) $\rightarrow$ x = y

Assume $\forall$z$\in$M (z $\in$ x $\iff$ z $\in$ y), we wish to show that x = y

Let z be arbitrarily chosen. ( Arguing from our Theory T)

As M is transitive, z $\in$ x $\iff$ z $\in$ M and z $\in$ x

therefore, by Assumption z $\in$ y

Similarly, for if z $\in$ y

Under our theory T, we have that two sets are equal iff they contain the same elements.

Thus, x = y.

Some Remarks:

We can think of relativization as saying that some formula is true in a Universe M, so that M "believes" that the formula is true. Only, considering elements of M, the formula is still true.

Our result gives us a sufficient condition for showing that a Class Satisfies the axiom of extentionality, we just need to show that our class is transitive. At least, if we are working within a Theory that has the Axiom of Extentionality.

Extentionality can fail to be true, if a Class isn't transitive, as some elements may be missing from the universe- and as a result it might believe that two sets share all of their elements.

Here is an example where Extentionality fails:

Consider the Universe A = {a,B,C} where B = {0,a} and C = {1,a},

It is easy to see that B is not equal to C. However, the only elements of A are a,B,C.

So, as far as A is concerned, the only element of B and C is a, and so they are equal, in the Universe A. A doesn't "see" that B contains 0, and C contains 1.

Final Remark

To show that a Theory T, proves M is a Model of ZFC.

We need to repeat this kind of argument for every axiom.