All Questions

Following Kunen's book, it makes clear that countable transitive models (ctm) exist only for a finite list of axioms of ZFC. So, why can we assume a ctm of the whole ZFC axioms exists and use it as ...

(1) Some typical inclompleteness proofs use a kind of fixed point argument - for certain $\Phi$ you find a $\varphi$ with $\Phi(\mathrm{code}(\varphi))\leftrightarrow\varphi$. (2) Some independence ...

Is it possible to prove $Con(ZFC) \rightarrow Con(ZFC + \neg CH)$ purely within ZFC? To prove this (using forcing) one seems to need a countable transitive model of ZFC. The texts I am reading avoid ...