# Models of ZFC and inaccessible cardinals

For a homework I have to show that working in $$ZFC$$, we have $$(ZC)^{V_{\lambda}}$$ for all limit ordinal $$\lambda>\omega$$.

Now in my lecture notes the relativization is defined for a class $$C$$ characterized by a formula $$\psi_C(x)$$ (i.e. $$x\in C$$ if and only if $$\psi_C(x)$$ holds) as follows : $$(\forall x\phi)^C:=\forall x(\psi_C(x)\rightarrow \phi^{C})$$ and $$(\exists x\phi)^C:=\exists x(\psi_C(x)\wedge \phi^C)$$ (etc...)

Which formula characterize the set $$V_{\lambda}$$ here in order to show $$ZFC\vdash (ZC)^{V_{\lambda}}$$? Or is there an other way to interpret the statement ?

• In general you need a parameter as well (e.g. $V_\lambda$ itself, or $\lambda$ if you want to be a bit more conservative) in your relativization. Commented May 17 at 17:56
• @NoahSchweber What do you mean by that ? How do you formalize it ? Commented May 17 at 18:02
• See my answer below. Commented May 17 at 18:11
• Where does the inaccessible from the title show up? Commented May 17 at 22:38

## 1 Answer

If $$\lambda$$ is an arbitrary limit ordinal there may not be a formula characterizing $$V_\lambda$$ as you want. However, we can get around this by considering relativization with parameters.

Suppose $$\varphi(x,y)$$ is a formula; we think of $$\varphi$$ as defining a class function, assigning to each object $$b$$ the class $$\{a: \varphi(a,b)\}$$. We can recursively define the relativization of an arbitrary formula $$\theta$$ (in which the variables $$x,y$$ don't occur, for simplicity) to a two-variable formula $$\varphi(x,y)$$ as follows:

• Booleans are boring (as in usual relativization).

• $$(\forall z\theta)^\varphi$$ is $$\forall z[\varphi(z,y)\rightarrow \theta^\varphi(z)].$$

• $$(\exists z\theta)^\varphi$$ is $$\exists z[\varphi(z,y)\wedge \theta^\varphi(z)]$$.

Crucially, you should convince yourself that if $$\theta$$ is a sentence then $$\theta^\varphi$$ is a formula with the single free variable $$y$$ and that we can interpret $$\theta^\varphi$$ as saying "$$\theta$$ holds in the class $$\{x: \theta(x,y)\}$$."

OK, now how does this pertain to your question?

Well, let $$\mu(x,y)$$ be the formula "($$y$$ is an ordinal and) $$x$$ is in the $$y$$th level of the cumulative hierarchy" and let $$\mathsf{BigLimOrd}(y)$$ be the formula "$$y$$ is a limit ordinal $$>\omega$$." Then your homework is asking you to prove the following in $$\mathsf{ZFC}$$:

For each axiom $$\alpha$$ of $$\mathsf{ZC}$$, we have $$\forall y[\mathsf{BigLimOrd}(y)\rightarrow\alpha^{\mu(x,y)}].$$

(Note that this is a bit more than just showing, for each axiom $$\alpha\in\mathsf{ZC}$$, that $$\mathsf{ZFC}\vdash\forall y[\mathsf{BigLimOrd}(y)\rightarrow\alpha^{\mu(x,y)}]$$; that said, if you can show that $$\forall y[\mathsf{BigLimOrd}(y)\rightarrow\mathsf{Powerset}^{\mu(x,y)}]$$ then you'll "morally" have completed most of the exercise.)

• Interesting, but I have then two questions ; 1) Why is there no problem anymore while writing "x is the $yth$ level of cumulative hierarchy" in the formula $\mu(x,y)$ ? Like, could we write it formally using the language of set theory (i.e the binary relation) ? 2) You say that showing $\Phi$ in $ZFC$ is "a bit more" than showing $ZFC\vdash \Phi$, but aren't those two statements the same ? Or, for you, does the first one mean something like : for all limit ordinal $y>\omega$, $V_{y}\models \alpha$, for $\alpha$ in $ZC$ ? Commented May 17 at 18:42
• @MamounAich For your first question: yes this is quite easily doable and a good exercise. For your second question, it's the difference between "For each $\alpha$, $\mathsf{ZFC}$ proves [thing about that specific $\alpha$]" and "$\mathsf{ZFC}$ proves the statement "For each appropriate $\alpha$, [thing]." For a more down-to-earth example of this sort of thing, recall that (i) for each $n$, $\mathsf{ZFC}$ proves "$n$ is not the code for a contradiction in $\mathsf{ZFC}$ but (ii) per Godel, $\mathsf{ZFC}$ doesn't prove "for each $n$, $n$ is not the code for a contradiction in $\mathsf{ZFC}$." Commented May 17 at 18:50
• Could you write me explicitly the formula $\mu(x,y)$ please ? I feel like we would have to define it recursively but it seems weird and I am not comfortable with those notions yet. Commented May 17 at 20:40
• @MamounAich Think about what properties the sequence of $V$-levels $$(V_\alpha)_{\alpha<y}$$ has (e.g. how long is it? what does it start with? how does it behave at successor levels? how does it behave at limit levels?). Commented May 17 at 21:41
• Thanks, however I already tried some formulas in order to define $\mu(x,y)$ in many ways but I'm not really sure of what I did. Can you write explicitly the definition of this formula down ? Commented May 17 at 21:55