I want to solve the problem "$V_{\omega}$ exists" can not be proved in $\textbf{ZC}$.
Where $\textbf{ZC}$ is the axiomatic set theory $\textbf{ZFC}$ except the axiom of replacement.
$V_0 = \emptyset, \quad V_{n+1}=P(V_n),\quad V_{\omega} = \bigcup_{i \in \omega} V_i$.
I know it is sufficent to construct a model $\mathcal{M}$ such that $\mathcal{M} \models \textbf{ZC}$ and $V_{\omega}$ is not in $\mathcal{M}$. Howerver, I can't add a infinite set to a small model or delete $V_{\omega}$ from a big model without breaking other axioms, especially the Seperation and the Power Set.
Any help. Thanks.
Best Answer
Here's a nice transitive model of ZC + "$V_\omega$ doesn't exist". I'll leave the proof to you.
Let $M_0=\omega,$ $M_{n+1}=P(M_n),$ and $M=\bigcup_{n<\omega} M_n.$