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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 ?

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  • $\begingroup$ 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. $\endgroup$ Commented May 17 at 17:56
  • $\begingroup$ @NoahSchweber What do you mean by that ? How do you formalize it ? $\endgroup$ Commented May 17 at 18:02
  • $\begingroup$ See my answer below. $\endgroup$ Commented May 17 at 18:11
  • $\begingroup$ Where does the inaccessible from the title show up? $\endgroup$
    – Asaf Karagila
    Commented May 17 at 22:38

1 Answer 1

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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.)

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  • $\begingroup$ 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 $ZC$ ? $\endgroup$ Commented May 17 at 18:42
  • $\begingroup$ @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 $\mathsf{ZFC}$." $\endgroup$ Commented May 17 at 18:50
  • $\begingroup$ 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. $\endgroup$ Commented May 17 at 20:40
  • $\begingroup$ @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?). $\endgroup$ Commented May 17 at 21:41
  • $\begingroup$ 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 ? $\endgroup$ Commented May 17 at 21:55

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