MOPA

I shall introduce a flexible new method showing that every countable model of PA admits a pointwise definable end-extension, one in which every individual is definable without parameters. And similarly for models of set theory, in which one may also achieve the Barwise extension result—every countable model of ZF admits a pointwise definable end-extension to a model of ZFC+V=L, or indeed any theory arising in a suitable inner model. A generalization of the method shows that every model of arithmetic of size at most continuum admits a Leibnizian extension, and similarly in set theory.

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This will be another talk in the MOPA series on the history of the subject.

The work on generalized quantifiers in formal systems of arithmetic was initiated in 1980 by Macintyre, motivated by the search for natural extensions of first-order arithmetic that are immune to the Kirby-Paris-Harrington style independence results. Some open questions posed by Macintyre were solved in a definitive way in 1982 by Schmerl and Simpson and after that Schmerl wrote two more papers on for Peano Arithmetic in the languages with Ramsay stationary quantifiers. Some results of Macintyre were obtained independently by Carl Morgenstern. All these papers, while very well written, are quite technical and not easily accessible for readers who are not familiar with more advanced tools of the model theory of arithmetic. I will survey the results suppressing most technical details. I will also talk about an attempt to use logic with stationary quantifiers to classify $\omega_1$-like recursively saturated models of PA.

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This will be another talk in the MOPA series on the history of the subject.

The work on generalized quantifiers in formal systems of arithmetic was initiated in 1980 by Macintyre, motivated by the search for natural extensions of first-order arithmetic that are immune to the Kirby-Paris-Harrington style independence results. Some open questions posed by Macintyre were solved in a definitive way in 1982 by Schmerl and Simpson and after that Schmerl wrote two more papers on for Peano Arithmetic in the languages with Ramsay stationary quantifiers. Some results of Macintyre were obtained independently by Carl Morgenstern. All these papers, while very well written, are quite technical and not easily accessible for readers who are not familiar with more advanced tools of the model theory of arithmetic. I will survey the results suppressing most technical details. I will also talk about an attempt to use logic with stationary quantifiers to classify $\omega_1$-like recursively saturated models of PA.

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Inspired by a certain result about PA in Albert Visser's paper 'Categories of theories and interpretations', I introduced the notions of tightness and solidity (of an arbitrary theory) in my paper 'Variations on a Visserian theme'; using them Visser's result can be expressed as: PA is a solid theory (it is easy to show that solidity implies tightness). My aforementioned paper demonstrates that besides PA, certain other canonical theories such as Z_2 (Second Order Arithmetic), ZF, and KM (Kelley-Morse Class Theory) are also solid. The first talk in this series will present : (a) the proofs of solidity of PA and Z_2, and (b) the relationship between Väänänen's notion of internal categoricity with the notions of solidity and tightness. The second part will concentrate on establishing the failure of solidity/tightness of certain subtheories of PA and Z_2, including any subtheory of PA or Z_2 that is finitely axiomatizable.

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Inspired by a certain result about PA in Albert Visser's paper 'Categories of theories and interpretations', I introduced the notions of tightness and solidity (of an arbitrary theory) in my paper 'Variations on a Visserian theme'; using them Visser's result can be expressed as: PA is a solid theory (it is easy to show that solidity implies tightness). My aforementioned paper demonstrates that besides PA, certain other canonical theories such as Z_2 (Second Order Arithmetic), ZF, and KM (Kelley-Morse Class Theory) are also solid. The first talk in this series will present : (a) the proofs of solidity of PA and Z_2, and (b) the relationship between Väänänen's notion of internal categoricity with the notions of solidity and tightness. The second part will concentrate on establishing the failure of solidity/tightness of certain subtheories of PA and Z_2, including any subtheory of PA or Z_2 that is finitely axiomatizable.

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Say that a theory $T$ is tight if any two distinct extensions of $T$ cannot be bi-interpretable. Vaguely speaking, tightness expresses a sort of maximality to the expressiveness of $T$. Visser showed that $\mathsf{PA}$ is tight and building on this work, Enayat showed that $\mathsf{Z}_2$, second-order arithmetic with full second-order comprehension, is also tight. In this talk I will address the question of whether full logical strength of these theories of arithmetic are necessary to have tightness, focusing on subsystems of $\mathsf{Z}_2$. The answer to this question is positive. If you restrict the comprehension axiom of $\mathsf{Z}_2$ to only arithmetical formulae, or if you restrict it to $\Sigma^1_k$ formulae, the resulting theory is not tight. As a specific instance, we show that if $M$ is either the minimum omega-model of $\mathsf{ACA}_0$ or the minimum beta-model of $\Pi^1_k$-$\mathsf{CA}$ for some $k \ge 1$, then $M$ is bi-interpretable with a carefully chosen extension $M[c] $ by Cohen-forcing.

This talk is about joint work with Alfredo Roque Freire.

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We construct a theory definitionally equivalent to first-order Peano arithmetic PA and a non-standard computable model of this theory. The same technique allows us to construct a theory definitionally equivalent to Zermelo-Fraenkel set theory ZF that has a computable model. See my preprint https://arxiv.org/abs/2209.00967 for more details.

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It is well known that every countable recursively saturated model of $\mathsf{PA}$ has a full compositional truth predicate; that is, such a model is expandable to the theory $\mathsf{CT}^-$. It is also well known that such a truth predicate need not be inductive, or indeed, need not satisfy even $\Delta_0$ induction. Recently, Enayat and Pakhomov showed that $\Delta_0$ induction for the truth predicate is equivalent to the principle of disjunctive correctness: the assertion that for any sequence of sentences $\langle \phi_i : i \lt c \rangle$, the disjunction $\bigvee\limits_{i \lt c} \phi_i$ is evaluated as true if and only if there is $i \lt c$ such that $\phi_i$ is evaluated as true. In the absence of $\Delta_0$ induction, various pathologies can occur, including models of $\mathsf{CT}^-$ for which all nonstandard length disjunctions are evaluated as true. In this talk, we classify the sets X for which there is a model of $\mathsf{CT}^-$ in which X is exactly the set of those c such that the disjunctions of length c of 0 = 1 is evaluated as false. In particular, we see that X can be $\omega$ if and only if $\omega$ is a strong cut, and therefore the 'disjunctively trivial' models mentioned before are in fact arithmetically saturated. This is joint work (in progress) with Mateusz Łełyk, drawing heavily on unpublished work by Jim Schmerl.

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