MOPA

We investigate the end extendibility of models of arithmetic with restricted elementarity. By utilizing the restricted ultrapower construction in the second-order context, for each $n \in \mathbb{N}$ and any countable model of $\mathrm{B}\Sigma_{n+2}$, we construct a proper $\Sigma_{n+2}$-elementary end extension satisfying $\mathrm{B}\Sigma_{n+1}$, which answers a question by Clote positively. We also give a characterization of countable models of $\mathrm{I}\Sigma_{n+2}$ in terms of their end extendibility similar to the case of $\mathrm{B}\Sigma_{n+2}$. Along the proof, we will introduce a new type of regularity principles in arithmetic called the weak regularity principle, which serves as a bridge between the model's end extendibility and the amount of induction or collection it satisfies.
The talk is based on this paper from arxiv:2409.03527.

Slides

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A theory is tight if and only if every two extensions of it, in the language of that theory, are bi-interpretable iff they are equal. The property of being tight can be seen as a kind of local categoricity in a suitable category of theories and interpretations. Examples of tight theories include $\text{PA}$, $\text{Z}_{2}$, $\text{ZF}$, and $\text{KM}$. Neatness, semantic tightness, and solidity are strengthenings of tightness, with solidity being the strongest and the other two being intermediate. During the talk we will focus on relations between those properties in the context of arithmetic theories and theories of finite sets.

Partly based on a joint work with Leszek Kołodziejczyk and Mateusz Łełyk.

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A theory is tight if and only if every two extensions of it, in the language of that theory, are bi-interpretable iff they are equal. The property of being tight can be seen as a kind of local categoricity in a suitable category of theories and interpretations. Examples of tight theories include $\text{PA}$, $\text{Z}_{2}$, $\text{ZF}$, and $\text{KM}$. Neatness, semantic tightness, and solidity are strengthenings of tightness, with solidity being the strongest and the other two being intermediate. During the talk we will focus on relations between those properties in the context of arithmetic theories and theories of finite sets.

Partly based on a joint work with Leszek Kołodziejczyk and Mateusz Łełyk.

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Over the last years, a lot of progress has been achieved in understanding the arithmetical strength of axiomatic theories of compositional truth. It turned out that a theory $\mathsf{CT}^-$ of compositional truth for arithmetical sentences can become non-conservative over $\mathsf{PA}$ upon adding some seemingly benign principles.

One of the principles whose arithmetical strength is still unknown is the axiom of propositional soundness which says that for any arithmetical sentence $\phi$ which is a propositional tautology, $\phi$ is true in the sense of the truth predicate. It is an open problem whether this axiom together with $CT^-$ is conservative over $PA$.

In our talk, we will show that if $(M,T)$ is a model of $\mathsf{CT}^-$ satisfying the propositional soundness principle, then $(M,T)$ satisfies a certain amount of saturation: if $(\phi_i)_{i \lt c}$ is a sequence of sentences such that for any standard $i$, $\phi_i$ is true in the sense of the truth predicate, then there is a nonstandard $d$ such that for each $i \in [0,d]$, $\phi_i$ is true. This puts very strong limitations on any possible conservativeness proof. The result may be seen as a counterpart to the classical theorem of Lachlan which says that the arithmetical part of any model of $\mathsf{CT}^-$ is recursively saturated.

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Let $\mathsf{M}$ be the weak set theory (with powersets) axiomatised by: $\textsf{Extensionality}$, $\textsf{Pair}$, $\textsf{Union}$, $\textsf{Infinity}$, $\textsf{Powerset}$, transitive containment ($\textsf{TCo}$), $\Delta_0\textsf{-Separation}$ and $\textsf{Set-Foundation}$. In this talk I will discuss the relationship between two alternative versions of the set-theoretic collection scheme: Collection and Strong Collection. Both of these schemes yield $\mathsf{ZF}$ when added to $\mathsf{M}$, but when restricted the $\Pi_n$-formulae (denoted $\Pi_n\textsf{-Collection}$ and $\textsf{Strong } \Pi_n\textsf{-Collection}$) these alternative versions of set-theoretic collection differ. In particular, over the theory $\mathsf{M}$, $\textsf{Strong }\Pi_n\textsf{-Collection}$ is equivalent to $\Pi_n\textsf{-Collection}+\Sigma_{n+1}\textsf{-Separation}$. And, $\mathsf{M}+\textsf{Strong }\Pi_n\textsf{-Collection}$ proves the consistency of $\mathsf{M}+\Pi_n\textsf{-Collection}$. In this talk I will show that, despite this difference in consistency strength, every countable well-founded model of $\mathsf{M}+\Pi_n\textsf{-Collection}$ satisfies $\textsf{Strong } \Pi_n\textsf{-Collection}$. If time permits I will outline how this argument can be refined to show that $\mathsf{M}+\Pi_n\textsf{-Collection}+\Pi_{n+1}\textsf{-Foundation}$ proves $\Sigma_{n+1}\textsf{-Separation}$.

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