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‎‎‎Note ‎to ‎the ‎following ‎well known theorem:‎

Theorem (1): ‎If ‎‎$‎‎\kappa‎‎$ ‎be a ‎‎"measurable" ‎cardinal ‎and ‎‎$‎‎‎\mathcal{F}‎$ be a‎ ‎"non-principal ‎$‎‎‎\kappa‎$-complete normal‎" ‎ultrafilter ‎on ‎it ‎then:‎‎ ‎‎‎‎$‎‎‎\langle V_{‎‎\kappa ‎+1‎},\in ‎‎\rangle ‎‎\cong ‎\prod_{‎\mathcal{F}‎}‎‎\lbrace ‎‎‎\langle ‎‎V_{‎\alpha +1‎}, \in \rangle‎~|~‎\alpha‎\in ‎\kappa‎ ‎\rbrace‎‎‎‎‎$ ‎‎‎‎

Proof: ‎Chang and‎ ‎Kiesler, ‎Model ‎Theory, ‎Page ‎241.‎ ‎

‎In ‎the ‎other ‎words ‎the ‎above ‎theorem ‎says ‎that ‎the ‎"truth" of a particular sentence ‎in ‎‎$‎‎‎\kappa +1‎$ th ‎level ‎of ‎von ‎Neumann's ‎cumulative hierarchy ‎is "‎dependent" ‎on ‎the ‎"truth" ‎of ‎that ‎sentence ‎in ‎lower ‎levels. ‎So ‎if a ‎sentence ‎be ‎"almost everywhere" ‎true ‎below ‎‎$‎‎‎\kappa ‎+1‎$ th ‎level, ‎it ‎cannot ‎be ‎false ‎in ‎‎it. In fact the lower stages give us a "truth approximation" for the ‎$‎‎‎\kappa +1‎$ th ‎stage. ‎Now ‎consider ‎the ‎following ‎definition: ‎‎ ‎

Definition (1): ‎We ‎say ‎that a‎ ‎cumulative ‎hierarchy ‎$‎W‎$‎ ‎has a ‎‎‎truth approximation property at ‎$‎‎‎\delta‎$ ‎th level (‎$‎‎tap(W,‎\delta‎)$‎) ‎iff ‎there exists an ultrafilter ‎$‎\mathcal{F}‎‎‎$ ‎on ‎$‎‎\delta‎‎$ ‎such that:‎‎ ‎‎‎‎‎‎‎$‎‎‎\langle W_{‎‎\delta +1‎‎},\in ‎‎\rangle ‎‎\cong ‎\prod_{‎\mathcal{F}‎}‎‎\lbrace ‎‎‎\langle W_{‎\alpha ‎+1‎}, \in \rangle‎~|~‎\alpha‎\in ‎\delta‎ ‎\rbrace‎‎‎‎‎$ ‎‎ ‎

Corollary (1): ‎$‎‎ZFC‎\Longrightarrow‎ \forall \kappa \in measurable~cardinal~~~tap(V,‎\kappa‎)$‎‎ ‎

Corollary (2): ‎$‎‎ZFC + ‎\exists~a~measurable~cardinal ‎\Longrightarrow‎ ‎\exists ‎‎\delta>‎\omega‎~~~tap(V,‎\delta‎)‎‎‎‎$‎ ‎

Now ‎there ‎are ‎some ‎natural ‎questions:‎

‎‎‎ Question (1): ‎Is ‎the ‎use ‎of non-trivial ‎‎$‎‎‎\kappa‎$-additive‎ normal measure ‎in ‎proof of ‎theorem ‎(1) ‎‎essential? ‎In ‎other ‎words ‎can ‎one ‎find a‎ ‎weaker ‎large ‎cardinal ‎axiom than existence of a measurable cardinal, ‎like ‎‎$‎A‎$ ‎such ‎tha‎t:‎ ‎

(a) ‎$‎‎ZFC+A‎\Longrightarrow ‎‎\exists ‎‎\delta‎>‎\omega‎~~~tap(V,‎\delta‎)‎‎$‎‎ ‎

Moreover can ‎‎‎$‎‎ZFC$ alone ‎prove ‎that ‎there is an ordinal ‎$‎\delta‎$‎ such that "‎$‎V‎$ ‎has a ‎truth approximation property at level ‎$‎\delta‎$‎‎‎"? Precisely ‎is ‎the ‎following ‎statement ‎true?‎ ‎

(b) ‎‎$‎‎Con(ZFC)‎\Longrightarrow ‎Con(ZFC+‎\forall ‎‎\delta‎>‎\omega‎~~~\neg tap(V,‎\delta‎)‎)‎$‎‎

Question (2): Is the inverse of corollary (2) true? In the other words does truth approximation property of von Neumann's cumulative hierarchy in a certain stage imply the existence of a measurable or weaker large cardinal? Precisely which one of these statements are true?

‎‎‎(a) $‎‎ZFC+‎\exists ‎‎\delta>‎\omega‎~~tap(V,‎\delta‎) ‎\Longrightarrow ‎‎\exists~‎a~strongly~inaccessible~cardinal‎‎$‎‎

(b) ‎$‎‎ZFC+‎\exists ‎‎\delta>‎\omega‎~~~tap(V,‎\delta‎) ‎\Longrightarrow ‎‎\exists~‎a~measurable~cardinal‎‎$ ‎

Question (3): ‎Are ‎there ‎any ‎known ‎"truth approximation properties" ‎for ‎other ‎famous ‎cumulative hierarchies ‎like ‎‎$‎‎L$ ‎and ‎‎$‎J‎$‎? Precisely is there any large cardinal axiom like ‎$‎‎A$ and ‎$‎B‎$‎ ‎which the following statements be true:‎

(a) ‎‎$‎‎ZFC+A‎\Longrightarrow ‎‎\exists‎‎\delta>‎\omega‎~~~tap(L,‎\delta‎)‎‎‎$‎

(b) ‎‎$‎‎ZFC+B‎\Longrightarrow ‎‎\exists‎‎\delta>‎\omega‎~~~tap(J,‎\delta‎)‎‎‎$‎ ‎‎‎‎ ‎

More simplified, are there any large cardinals ‎$‎‎‎\kappa‎$ ‎and ‎‎$‎‎\lambda‎‎$ and ultrafilters ‎$‎‎\mathcal{F}‎$ ‎and ‎$‎‎‎\mathcal{G}‎$ ‎on them ‎‎‎‎such ‎that the following statements be true?‎ ‎

(c)‎‎ ‎‎‎‎‎‎‎$‎‎‎\langle L_{‎‎‎\kappa‎‎ ‎+1‎},\in ‎‎\rangle ‎‎\cong ‎\prod_{‎\mathcal{F}‎}‎‎\lbrace ‎‎‎\langle L_{‎\alpha +1‎}, \in \rangle‎~|~‎\alpha‎\in ‎\kappa‎ ‎\rbrace‎‎‎‎‎$ ‎‎ ‎

(d) ‎‎‎‎‎‎‎$‎‎‎\langle J_{‎‎‎\lambda‎‎ ‎+1‎},\in ‎‎\rangle ‎‎\cong ‎\prod_{‎\mathcal{G}‎}‎‎\lbrace ‎‎‎\langle J_{‎\alpha +1‎}, \in \rangle‎~|~‎\alpha‎\in ‎‎\lambda‎‎ ‎\rbrace‎‎‎‎‎$ ‎‎

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up vote 4 down vote accepted

For question (1), the answer is the truth approximation property at $\delta$ implies the existence of a measurable cardinal. This is simply because the filter $\mathcal{F}$ witnessing your isomorphism must be countably complete, or else the ultraproduct on the right hand side will have an ill-founded $\omega$, preventing the isomorphism. But if an ultrafilter is countably complete, then the $\kappa$ for which it is $\kappa$-complete but not $\kappa^+$-complete is a measurable cardinal. That is, the degree of completeness of any countably complete ultrafilter is a measurable cardinal.

Thus, the converse of corollary 2 is true, and this answers question 2.

For question 3, you can take A and B to be "there is a measurable cardinal". The way I think about it is this. If $\delta$ is a measurable cardinal, with normal measure $\mathcal{F}$, then $\kappa$ is represented in the ultrapower $j:V\to M$ by the function $\alpha\mapsto\alpha$, and so $\kappa+1$ is represented by the function $\alpha\mapsto\alpha+1$. Thus, the $L_{\delta+1}$ of $M$, which is the real $L_{\delta+1}$, is represented by the function $\alpha\mapsto L_{\alpha+1}$, which gives your property (c). And similarly with $J_{\delta+1}$ and $\alpha\mapsto J_{\alpha+1}$, which gives (d).

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