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Hanf numbers for extendibility and related phenomena

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Abstract

This paper contains portions of Baldwin’s talk at the Set Theory and Model Theory Conference (Institute for Research in Fundamental Sciences, Tehran, October 2015) and a detailed proof that in a suitable extension of ZFC, there is a complete sentence of \(L_{\omega _1,\omega }\) that has maximal models in cardinals cofinal in the first measurable cardinal and, of course, never again.

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Notes

  1. We say \({\varvec{K}}\) is universally extendible in \(\kappa \) if \(M \in {\varvec{K}}\) with \(|M| =\kappa \) has a proper \(\prec _{{\varvec{K}}}\)-extension in the class. Here, this means has an \(\infty ,\omega \)-elementary extension.

  2. Due to our tardiness in preparing this paper it could not be included in the special volume dedicated to the 2015 conference.

  3. Inductively, Hjorth shows at each \(\alpha \) and each member \(\phi \) of \(S_\alpha \) one of two sentences, \(\chi _\phi , \chi '_\phi \), works as \(\phi _{\alpha +1}\) for \(\aleph _{\alpha +1}\).

  4. A trivial term (or polynomial) is one which is identically 0.

  5. Called strong in [12].

  6. Equivalently for Boolean algebras, if every non-zero element is above at least one atom.The conditions are not equivalent on an arbitrary distributive lattice.

  7. The subsets of \(P^M_0\) are not elements of M.

  8. But \(b^*\) is not constant in the vocabulary; as the models are extended, \(b^*\) changes.

  9. A further equivalence: \(|Atom(B_{n_*})| - |P^M_{4,1}|\) is a power of two.

  10. As in Definition 3.1.1 with \(X= \emptyset \).

  11. Clearly, this could be achieved by choosing a new copy of \(\mathbb {B}_1\).

  12. Abusing notation, since \(\mathbb {B}_1\) is not a \(\tau \)-structure, we write \(P^{\mathbb {B}_1}_{4,1}\) for the set of atoms of \(\mathbb {B}_1\) and \(P^{\mathbb {B}_1}_{4}\) for their finite joins.

  13. \(\mathbb {B}'_2\) is freely generated as a Boolean algebra by (isomorphic copies of) \(\mathbb {B}_1\) and \(P^{N_2}_1\) over \(\mathbb {B}^*\).

  14. This is the crucial application of Lemma 3.1.10 which stengthened our notion of independence by getting a standard consequence of exchange, even though exchange fails here.

  15. For local intelligibility (and at the risk of global confusion) we use indices \(b_{\gamma }\) and \(M_{\gamma }\) rather than \(b_{\alpha _\gamma }\) and \(M_{\alpha _\gamma }\) that would keep more precise track of the subsequence fact.

  16. Shelah’s theory of excellence concerns unique free disjoint amalgamations of infinite structures in \(\omega \)-stable classes of models of complete sentences in \(L_{\omega _1,\omega }\).

  17. We say amalgamation holds in \(\kappa \) in the trivial special case when all models in \(\kappa \) are maximal.

    We say amalgamation fails in \(\kappa \) if there are no models to amalgamate.

  18. Kueker, as reported in [24], gave the first example of a complete sentence failing amalgamation in \(\aleph _0\).

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Authors and Affiliations

  1. University of Illinois at Chicago, Chicago, USA

    John T. Baldwin

  2. Hebrew University of Jerusalem, Jerusalem, Israel

    Saharon Shelah

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  1. John T. Baldwin
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John T. Baldwin: Research partially supported by Simons travel Grant G5402, G3535. Saharon Shelah: Item 1092 on Shelah’s publication list. Partially supported by European Research Council Grant 338821, and by National Science Foundation Grant 136974.

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Baldwin, J.T., Shelah, S. Hanf numbers for extendibility and related phenomena. Arch. Math. Logic 61, 437–464 (2022). https://doi.org/10.1007/s00153-021-00796-1

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