Shelah's conjecture fails for higher cardinalities

Abstract

The main goal of this paper is to generalize the results that where presented in [11] for $\aleph_1$-Kurepa trees to $\aleph_{\alpha+1}$-Kurepa trees. We construct an $\mathcal{L}_{\omega_1,\omega}$-sentence $\psi_{\alpha}$, that codes $\aleph_{\alpha+1}$-Kurepa trees, for some countable $\alpha$. One of the main results for its spectrum is the following: It is consistent that $2^{\aleph_{\alpha}}<2^{\aleph_{\alpha+1}}$, that $2^{\aleph_{\alpha+1}}$ is weakly inaccessible and that the spectrum of $\psi_{\alpha}$ is equal to $[\aleph_0, 2^{\aleph_{\alpha+1}})$. This relates to a conjecture of Shelah, that if $\aleph_{\omega_1}<2^{\aleph_0}$ and there is a model of some $\mathcal{L}_{\omega_1,\omega}$-sentence of size $\aleph_{\omega_1}$, then there is a model of size $2^{\aleph_0}$. Shelah calls $\aleph_{\omega_1}$ 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 $2^{\aleph_{\alpha+1}}$, it is consistent that there is no local Hanf number. There are some interesting results for the amalgamation spectrum too. We prove that $\kappa$-amalgamation for $\mathcal{L}_{\omega_1,\omega}$-sentences is not absolute. More specifically we prove for $\alpha>0$ finite, it is consistent that: 1) $2^{\aleph_{\alpha}} = \aleph_{\alpha+1}<\lambda \leq 2^{\aleph_{\alpha+1}}, cf(\lambda)>\aleph_{\alpha}$ and $AP-Spec(\psi_{\alpha})$ contains the whole interval $[\aleph_{\alpha+2}, \lambda]$ and possibly $\aleph_{\alpha+1}$. 2) $2^{\aleph_{\alpha}} = \aleph_{\alpha+1}<2^{\aleph_{\alpha+1}}$, $2^{\aleph_{\alpha+1}}$ is weakly inaccessible and $AP-Spec(\psi_{\alpha})$ contains the whole interval $[\aleph_{\alpha + 2}, 2^{\aleph_{\alpha+1}})$ and possibly $\aleph_{\alpha+1}$.