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?
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\} $
The collection of formulas is countable.
Based on 1) and 2) I conclude: there are only a countable number of classes.
Every set is a class.
Based on 3) and 4) I conclude: there are only a countable number of sets.