Here's a question I can't answer by myself: The Reflection Principle in Set Theory states for each formula $\phi(v\_{1},...,v\_{n})$ and for each set M there exists a set N which extends M such that the following holds


$\phi^{N} (x\_{1},...,x\_{n})$ iff $\phi (x\_{1},...,x\_{n})$ for all $x\_{1},...x\_{n} \in N$


Thus if $\sigma$ is a true sentence then the RFP yields a model of it and as a consequence any finite set of axioms of ZFC has a model (as a consequence ZFC is not finitely axiomatizable by Gödel's Second Incompleteness Theorem)


But why can't I just use now the Compactness Theorem (stating that each infinte set of formulas such that each finite subset has a model, has a model itself) to obtain a model of ZFC (which is actually impossible)??



Montague's Reflection Principle and Compactness Theorem
For any finite set of axioms K of ZFC, ZFC proves "K has a model", via the reflection principle as you note. However, ZFC does not prove "for any finite set of axioms K of ZFC, K has a model". The distinction between these two is what prevents ZFC from proving that ZFC has a model.


(That is, even though, as you note, ZFC proves "if every finite set of axioms K of ZFC has a model, then ZFC has a model", as ZFC proves compactness, it does not follow that ZFC proves the consequent of this implication, as in fact ZFC does not prove the antecedent; ZFC only proves each particular instance of the antecedent, but not the universal statement itself.)


