Mathematics – Logic
Scientific paper
2010-07-14
Mathematics
Logic
26 pages Submitted for publication- 2010 07 14
Scientific paper
We study which cardinals are characterizable by a Scott sentence, in the sense that $\phi_M$ characterizes $\kappa$, if it has a model of size $\kappa$, but not of $\kappa^+$. We show that if $\aleph_\alpha$ is characterizable by a Scott sentence and $\beta<\omega_1$, then $\aleph_{\alpha+\beta}$ is characterizable by a Scott sentence. Under the same assumption, $\aleph_{\alpha}^{\aleph_0}$ is also characterizable by a Scott sentence and if $0<\gamma<\omega_1$, then the same is true for $2^{\aleph_{\alpha+\gamma}}$. Characterizable cardinals are closed under countable unions and products. As a corollary we get that for countable $\alpha, \beta$, $\aleph_{\alpha}^{\aleph_{\beta}}$ is characterizable by a Scott sentence. Following work of Baumgartner and Malitz it is natural to consider when a cardinal can be homogeneously characterized by a Scott sentence. Many of the issues remain profoundly unclear, however, we do prove that if $\kappa$ is characterizable, then $2^{(\kappa^+)}$ is homogeneously characterizable.
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