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 Goedel'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)??

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)??