Where am I wrong in this "proof" that the collection of sets is countable?

Question

Looking at sets (in ZF, Set Theory of Kunen) I could not escape from looking at logic as well.

A formal language is presented containing basic symbols ($\wedge,\neg,\exists,(,),\in,=,$ and $v_{i}$ for $i=0,1,\dots$).

Also things are said about free variables and bound variables.

Formulas are defined inductively ($v_{i}=v_{j}$ and $v_{i}\in v_{j}$ for any $i,j\in\left\{ 0,1,\dots\right\} $ and if $\phi,\psi$ are formulas then so are $\left(\phi\right)\wedge\left(\psi\right),\neg\left(\phi\right),\exists v_{i}\left(\phi\right)$for every $i\in\left\{ 0,1,\dots\right\} $).

Now my question: where am I wrong in the following statements/reasoning?

1. For every class $K$ there is a formula $\phi$ containing a free variable $v_{0}$ such that $K=\left\{ v_{0}\mid\phi\left(v_{0}\right)\right\} $

2. The collection of formulas is countable.

3. Based on 1) and 2) I conclude: there are only a countable number of classes.

4. Every set is a class.

5. Based on 3) and 4) I conclude: there are only a countable number of sets.

Answer 1

There are two points here:

1. The enumeration of the formulas don't live "inside the model", it lives "in the meta-theory". It is something that the model has access to. And indeed, a model where every set is definable without parameters is necessarily countable in the meta-universe.

2. You are forgetting parameters. Every set is a class, that is true. But it is possible that there are sets that only have "nearly trivial definition", in the sense that you can't really defined them in terms of "simpler" sets. So such set is ostensibly defined by the formula $x\in p$, where $p$ is a parameter and you place the set itself as the parameter.

   That is not a "true" definition in the sense that we think of, but it is legal in the sense that when a model is given you want to define a class and you are allowed to use parameters from the model, in particular you can use the set itself as a parameter when you define a class. So you define a class and it happens to be exactly the set from the parameter.

You are correct, however, that at most countably many sets are definable without parameters over any model of $\sf ZFC$. But they are definable externally to the model. And the enumeration is also external to the model.