Shelah's conjecture fails for higher cardinalities

The main goal of this paper is to generalize the results that where presented in [11] for ${\mathrm{\aleph }}_1$-Kurepa trees to ${\mathrm{\aleph }}_{\alpha +1}$-Kurepa trees. We construct an ${\mathcal{L}}_{{\omega }_1,\omega }$-sentence ${\psi }_{\alpha }$, that codes ${\mathrm{\aleph }}_{\alpha +1}$-Kurepa trees, for some countable $\alpha$. One of the main results for its spectrum is the following: It is consistent that $2^{{\mathrm{\aleph }}_{\alpha }}<2^{{\mathrm{\aleph }}_{\alpha +1}}$, that $2^{{\mathrm{\aleph }}_{\alpha +1}}$ is weakly inaccessible and that the spectrum of ${\psi }_{\alpha }$ is equal to $[{\mathrm{\aleph }}_0,2^{{\mathrm{\aleph }}_{\alpha +1}})$. This relates to a conjecture of Shelah, that if ${\mathrm{\aleph }}_{{\omega }_1}<2^{{\mathrm{\aleph }}_0}$ and there is a model of some ${\mathcal{L}}_{{\omega }_1,\omega }$-sentence of size ${\mathrm{\aleph }}_{{\omega }_1}$, then there is a model of size $2^{{\mathrm{\aleph }}_0}$. Shelah calls ${\mathrm{\aleph }}_{{\omega }_1}$ the local Hanf number below $2^{{\mathrm{\aleph }}_0}$ 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 $2^{{\mathrm{\aleph }}_0}$ by $2^{{\mathrm{\aleph }}_{\alpha +1}}$, 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) $2^{{\mathrm{\aleph }}_{\alpha }}={\mathrm{\aleph }}_{\alpha +1}<\lambda \le 2^{{\mathrm{\aleph }}_{\alpha +1}},cf(\lambda )>{\mathrm{\aleph }}_{\alpha }$ and $AP-Spec({\psi }_{\alpha })$ contains the whole interval $[{\mathrm{\aleph }}_{\alpha +2},\lambda ]$ and possibly ${\mathrm{\aleph }}_{\alpha +1}$. 2) $2^{{\mathrm{\aleph }}_{\alpha }}={\mathrm{\aleph }}_{\alpha +1}<2^{{\mathrm{\aleph }}_{\alpha +1}}$, $2^{{\mathrm{\aleph }}_{\alpha +1}}$ is weakly inaccessible and $AP-Spec({\psi }_{\alpha })$ contains the whole interval $[{\mathrm{\aleph }}_{\alpha +2},2^{{\mathrm{\aleph }}_{\alpha +1}})$ and possibly ${\mathrm{\aleph }}_{\alpha +1}$.