Shelah's conjecture fails for higher cardinalities
Georgios Marangelis
Abstract
The main goal of this paper is to generalize the results that where presented in [11] for $\aleph_1$-Kurepa trees to $\aleph_{α+1}$-Kurepa trees. We construct an $\mathcal{L}_{ω_1,ω}$-sentence $ψ_α$, that codes $\aleph_{α+1}$-Kurepa trees, for some countable $α$. One of the main results for its spectrum is the following: It is consistent that $2^{\aleph_α}<2^{\aleph_{α+1}}$, that $2^{\aleph_{α+1}}$ is weakly inaccessible and that the spectrum of $ψ_α$ is equal to $[\aleph_0, 2^{\aleph_{α+1}})$. This relates to a conjecture of Shelah, that if $\aleph_{ω_1}<2^{\aleph_0}$ and there is a model of some $\mathcal{L}_{ω_1,ω}$-sentence of size $\aleph_{ω_1}$, then there is a model of size $2^{\aleph_0}$. Shelah calls $\aleph_{ω_1}$ the local Hanf number below $2^{\aleph_0}$ and proves the consistency of his conjecture in [9]. It is open if the negation of Shelah's conjecture is consistent. Our result proves that if we replace $2^{\aleph_0}$ by $2^{\aleph_{α+1}}$, it is consistent that there is no local Hanf number. There are some interesting results for the amalgamation spectrum too. We prove that $κ$-amalgamation for $\mathcal{L}_{ω_1,ω}$-sentences is not absolute. More specifically we prove for $α>0$ finite, it is consistent that: 1) $2^{\aleph_α} = \aleph_{α+1}<λ\leq 2^{\aleph_{α+1}}, cf(λ)>\aleph_α$ and $AP-Spec(ψ_α)$ contains the whole interval $[\aleph_{α+2}, λ]$ and possibly $\aleph_{α+1}$. 2) $2^{\aleph_α} = \aleph_{α+1}<2^{\aleph_{α+1}}$, $2^{\aleph_{α+1}}$ is weakly inaccessible and $AP-Spec(ψ_α)$ contains the whole interval $[\aleph_{α+ 2}, 2^{\aleph_{α+1}})$ and possibly $\aleph_{α+1}$.