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Kurepa trees and spectra of $\mathcal{L}_{\omega_1,\omega}$-sentences. (arXiv:1705.05821v1 [math.LO])
来源于:arXiv
We construct a \emph{single} $\mathcal{L}_{\omega_1,\omega}$-sentence $\psi$
that codes Kurepa trees to prove the consistency of the following:
(1) The spectrum of $\psi$ is consistently equal to
$[\aleph_0,\aleph_{\omega_1}]$ and also consistently equal to
$[\aleph_0,2^{\aleph_1})$, where $2^{\aleph_1}$ is weakly inaccessible.
(2) The amalgamation spectrum of $\psi$ is consistently equal to
$[\aleph_1,\aleph_{\omega_1}]$ and $[\aleph_1,2^{\aleph_1})$, where again
$2^{\aleph_1}$ is weakly inaccessible.
This is the first example of an $\mathcal{L}_{\omega_1,\omega}$--sentence
whose spectrum and amalgamation spectrum are consistently both right-open and
right-closed. It also provides a positive answer to a question in [14].
(3) Consistently, $\psi$ has maximal models in finite, countable, and
uncountable many cardinalities. This complements the examples given in [1] and
[2] of sentences with maximal models in countably many cardinalities.
(4) $2^{\aleph_0}<\aleph_{\omega_1}<2^{\ale 查看全文>>