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Second order arithmetic as the model companion of set theory

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Abstract

This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin’s generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a \(\Pi _2\)-property formalized in an appropriate language for second order number theory is forcible from some \(T\supseteq \mathsf {ZFC}+\)large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T. In particular we show that the first order theory of \(H_{\omega _1}\) is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for \(\Delta _0\)-properties and for all universally Baire sets of reals. We will extend these results also to the theory of \(H_{\aleph _2}\) in a follow up of this paper.

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Notes

  1. It is a standard result of set theory that \(\Delta _0\)-formulae define absolute properties for transitive models of \(\mathsf {ZFC}\). On the other hand the notion of universal Baireness captures exactly those sets of reals whose first order properties cannot be changed by means of forcing (for example all Borel sets of reals are universally Baire). Therefore these predicates have a meaning which is clear across the different models of set theory. See also the last part of this introduction.

  2. On a more conceptual note, this came as the realisation of a conceptual similarity between the search of arbitrary models for set theory and those displaying generic absoluteness results. See [18] for a discussion on this point.

  3. These results will also count as a justification of the use of large cardinals in the study of the models of set theory. See [5] and [27] for a thorough discussion.

  4. A much more extensive analysis of which signatures for set theory can be considered worth of attention and bring model companionship results for extensions of \(\mathsf {ZFC}+\)large cardinals is carried on in a follow up of this paper [24] by the second author.

  5. All problems of second order arithmetic are first order properties of \(H_{\omega _1}\).

  6. For example let T be the theory of commutative rings with no zero divisors which are not fields. Then \({\mathcal {E}}_T\) is exactly the class of algebraically closed fields and no model in \({\mathcal {E}}_T\) is a model of T.

  7. For a general overview of the many multiverse conceptions in the literature see [1, 17], and [14].

  8. For example let \({\dot{R}}\) be a canonical name for a binary relation on \(\omega \) coding the transitive closure of \(\lbrace \sigma _G \rbrace \) for any G V-generic for \(\mathsf {B}\) with \(b\in G\). Then for any such G the transitive collapse of \({\dot{R}}_G\) is the transitive closure of \(\lbrace \sigma _G \rbrace \), and clearly \({\dot{R}}\) is decided by countably many antichains below b.

  9. There can be morphisms \(h:H_\kappa ^{\mathsf {B}}/_G\rightarrow H_\delta ^{\mathsf {C}}/_H\) which are not of the form \({\hat{f}}/_H\) for some complete homomorphism \(f:\mathsf {B}\rightarrow \mathsf {C}\), even in case \(\mathsf {B}\) preserve the regularity of \(\kappa \) and \(\mathsf {C}\) the regularity of \(\delta \). We do not spell out the details of such possibilities.

  10. See [9,  Section 25] and in particular the statement and proof of Lemma 25.25, which contains all ideas on which one can elaborate to draw the conclusions below.

  11. A quantifier free \(\tau _{A_1,\dots ,A_k}\)-formula is a boolean combination of atomic \(\tau _{{{\,\mathrm{\mathsf {ST}}\,}}}\)-formulae with formulae of type \(A_j(\vec {x})\). For example \(\exists x\in y A(y)\) is not a quantifier free \(\tau _{{{\,\mathrm{\mathsf {ST}}\,}}}\)-formula, and is actually equivalent to the \(\Sigma _1\)-formula \(\exists x(x\in y)\wedge A(y)\).

  12. The assumptions are an overkill: class many Woodin cardinals suffice. However we write the stronger hypothesis so to be able to quote verbatim the relevant results as proved in [12].

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Authors and Affiliations

  1. Universidade Estadual de Campinas, Campinas, Brazil

    Giorgio Venturi

  2. Università degli studi di Torino, Torino, Italy

    Matteo Viale

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  1. Giorgio Venturi
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  2. Matteo Viale
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Correspondence to Giorgio Venturi.

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The first author acknowledges support from the FAPESP Jovem Pesquisador Grant n. 2016/25891-3. The second author acknowledges support from INDAM through GNSAGA and from the project: PRIN 2017-2017NWTM8R Mathematical Logic: models, sets, computability. We thank the referee for the careful work of revision and several useful comments.

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Venturi, G., Viale, M. Second order arithmetic as the model companion of set theory. Arch. Math. Logic 62, 29–53 (2023). https://doi.org/10.1007/s00153-022-00831-9

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