TITLE: cardinality of math theorems
QUESTION [5 upvotes]: Some naive questions from an interested layman regarding the cardinality of the set of all math theorems (discovered and undiscovered).
1) What branches of math are not contained in ZF set theory + the axiom of choice + the continuum hypothesis?
2) What is the cardinality of all discovered and undiscovered theorms under ZF set theory?
3) Cardinality of theorems under ZF + axiom of choice?
4) Cardinality of theorems under ZF + continuum?
5) Cardinality of theorems under ZF + axiom of choice + continuum?

REPLY [12 votes]: Each of these theories has a countable infinity of theorems. It is immediate that they have at least that many theorems, because $n=n$ is a theorem of each of them for any numeral $n$. On the other hand, every theorem can, by definition, be represented as a finite sequence of symbols from a countable (or finite) alphabet, and there are only countably many such sequences.