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