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Non-absoluteness of model existence at $\aleph _\omega $

Tom 243 / 2018

David Milovich, Ioannis Souldatos Fundamenta Mathematicae 243 (2018), 179-193 MSC: Primary 03C48, 03C55; Secondary 03E35, 03E55, 03C52, 03C75. DOI: 10.4064/fm419-12-2017 Opublikowany online: 25 June 2018

Streszczenie

Friedman et al. (2013) considered the question whether model existence of $\mathcal{L}_{\omega_1,\omega}$-sentences is absolute for transitive models of ZFC, in the sense that if $V \subseteq W$ are transitive models of ZFC with the same ordinals, $\varphi\!\in\! V$ and $V\!\models\! “\varphi \text{ is an } \mathcal{L}_{\omega_1,\omega}\text{-sentence}”$, then $V \models\Phi$ if and only if $W \models\Phi$ where $\Phi$ is a first-order sentence with parameters $\varphi$ and $\alpha$ asserting that $\varphi$ has a model of size $\aleph_\alpha$.

From Friedman et al. (2013) we know that the answer is positive for $\alpha=0,1$, and under the negation of CH the answer is negative for all $\alpha \gt 1$. Under GCH, and assuming the consistency of a supercompact cardinal, the answer remains negative for each $\alpha \gt 1$, except the case when $\alpha=\omega$ which is an open question in Friedman et al. (2013).

We answer the open question by providing a negative answer under GCH even for $\alpha=\omega$. Our examples are incomplete sentences. In fact, the same sentences can be used to prove a negative answer under GCH for all $\alpha \gt 1$ assuming the consistency of a Mahlo cardinal. Thus, the large cardinal assumption is relaxed from a supercompact in Friedman et al. (2013) to a Mahlo cardinal.

Finally, we consider the absoluteness question for the $\aleph_\alpha$-amalgamation property of $\mathcal{L}_{\omega_1,\omega}$-sentences (under substructure). We prove that assuming GCH, $\aleph_\alpha$-amalgamation is non-absolute for $1 \lt \alpha \lt \omega$. This answers a question of Sinapova and Souldatos (2017). The cases $\alpha=1$ and $\alpha$ infinite remain open. As a corollary we show that it is non-absolute that the amalgamation spectrum of an $\mathcal{L}_{\omega_1,\omega}$-sentence is empty.

Autorzy

  • David MilovichDepartment of Mathematics and Physics
    Texas A&M International University
    5201 University Blvd.
    Laredo, TX 78045, U.S.A.
    e-mail
  • Ioannis SouldatosMathematics Department
    University of Detroit Mercy
    4001 W. McNichols
    Detroit, MI 48221, U.S.A.
    e-mail

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