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Shelah's main gap theorem in model theory says:

For each first order complete theory $T$ in a countable language if $I(T,\kappa)$ denotes the number of its models of size $\kappa$ then one of these situations holds:

(a) $\forall \alpha>0~~~~~I(T,\aleph_\alpha)=2^{\aleph_\alpha}$

(b) $\forall \alpha>0~~~~~I(T,\aleph_\alpha)<\beth_{\omega_{1}}(|\alpha|)$

Now consider the open interval $(\beth_{\omega_{1}}(|\alpha|),2^{\aleph_\alpha})$ for each given positive ordinal $\alpha$. I would like to know how large or small can this gap be, up to consistency? In the other words, what is the "diameter" of the "gap-zone" $(\beth_{\omega_{1}}(|\alpha|),2^{\aleph_\alpha})$ in Shelah's theorem and how this diameter does change by increasing cardinals? Precisely:

Question: Let $F:Ord\setminus\{0\}\longrightarrow Card$ be as follows: $$\forall\alpha>0~~~~~F(\alpha):=|(\beth_{\omega_{1}}(|\alpha|),2^{\aleph_\alpha})\cap Card|$$

(a) What is the minimum definable cardinal $\lambda$ such that:

$$Con(ZFC+~\text{Possibly some extra assumptions}~)\Longrightarrow Con(ZFC+\forall\alpha\in Ord^{>0}~~~~~F(\alpha)\leq \lambda)$$

(Assuming GCH we can choose $\lambda=0$)

(b) For which definable cardinal $\lambda$ we have:

$$Con(ZFC+~\text{Possibly some extra assumptions}~)\Longrightarrow Con(ZFC+\forall\alpha\in Ord^{>0}~~~~~F(\alpha)\geq \lambda)$$

How large can such a cardinal be?

Remark: Any other criterion for measuring the largeness of the gap zone $(\beth_{\omega_{1}}(|\alpha|),2^{\aleph_\alpha})$ is welcome. I think the exponential nature of the lower bound in Shelah's theorem adds some limitations on the largeness of this gap.

Motivation: It is possible to prove theorems which give us different amount of information up to consistency, depends on the various situations of the parameters in a particular model of ZFC with special properties. For example in Shelah's main gap theorem the quantity $2^{\aleph_\alpha}$ has no determined value by alephs and this can change the largeness of the gap which Shelah's theorem describes.

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    $\begingroup$ Gaps are usually measured in order type, not cardinality. Otherwise, how can you differ a gap of $\omega$ and a gap of $\omega^3+1$? $\endgroup$ – Asaf Karagila Jun 7 '14 at 14:20
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    $\begingroup$ Gap, not gape. $\endgroup$ – Andrés E. Caicedo Jun 8 '14 at 0:36
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    $\begingroup$ Just as a remark, in question 2 necessarily $\lambda=0$, since your interval is empty whenever $\alpha$ is an $\aleph$-fixed point. $\endgroup$ – Miha Habič Jun 8 '14 at 21:10

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