MOPA

The disjunctive correctness principle (DC) states that a disjunction of arbitrary (possibly nonstandard) length is true if and only if one of its disjuncts is true. On first sight, the principle seems an innocent and natural generalization of the familiar compositional truth axiom for disjunction. However, Ali Enayat and Fedor Pakhomov demonstrated that (DC) has the same strength as Delta_0 induction, hence it produces a non-conservative extension of the background arithmetical theory.

In the presentation the proof of a stronger result will be presented. Let (DC-Elim) be just one direction of (DC), namely, the implication 'if a disjunction is true, then one of it disjuncts is true'. We will show that already (DC-Elim) carries the full strength of Delta_0 induction; moreover, the proof of this fact will be significantly simpler than the original argument of Enayat and Pakhomov.

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This is joint work (still in progress) with Mateusz Łełyk (who gave the first part of the talk). By an axiomatic theory of truth (for the language of arithmetic, $L$) we mean a theory in L enriched with a fresh unary predicate $T(x)$ which (extends the elementary arithmetic EA and) proves all sentences of the form ($\phi$ being a sentence in L) $T(\phi)\equiv \phi.$

The collection of all sentence of the above form is normally called $TB^-$. It is well known that axiomatic theories of truth have a number of interesting model-theoretic consequences. For example, already relatively weak theories of truth impose recursive saturation, in the sense that the L-reduct of any model of such theory is recursively saturated. To give another example, already $TB^-$ imposes elementary equivalence of models, in the sense that whenever $(M,T)\models TB^-$, $(M',T')\models TB^-$, and $(M,T)\subset (M', T)$ (the first model is a submodel of the second one), then actually $M$ and $M'$ are elementarily equivalent. During (both parts) of the talk we investigate which of these properties actually characterize the respective truth theory up to definability. In particular, in the first part of the talk, we prove the following results (we restrict ourselves to theories in a finite language and extending EA):

1. Every theory which imposes elementary equivalence defines $TB^-$.
2. Every theory which imposes full elementarity defines $UTB^-$.

Additionally, we take a look at the definability relations between axiomatic truth theories and axiomatic theories of definability or skolem functions.

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This is joint work (still in progress) with Bartosz Wcisło (who will give the second part of the talk). By an axiomatic theory of truth (for the language of arithmetic, $L$) we mean a theory in L enriched with a fresh unary predicate $T(x)$ which (extends the elementary arithmetic EA and) proves all sentences of the form ($\phi$ being a sentence in L) $T(\phi)\equiv \phi.$

The collection of all sentence of the above form is normally called $TB^-$. It is well known that axiomatic theories of truth have a number of interesting model-theoretic consequences. For example, already relatively weak theories of truth impose recursive saturation, in the sense that the L-reduct of any model of such theory is recursively saturated. To give another example, already $TB^-$ imposes elementary equivalence of models, in the sense that whenever $(M,T)\models TB^-$, $(M',T')\models TB^-$, and $(M,T)\subset (M', T)$ (the first model is a submodel of the second one), then actually $M$ and $M'$ are elementarily equivalent. During (both parts) of the talk we investigate which of these properties actually characterize the respective truth theory up to definability. In particular, in the first part of the talk, we prove the following results (we restrict ourselves to theories in a finite language and extending EA):

1. Every theory which imposes elementary equivalence defines $TB^-$.
2. Every theory which imposes full elementarity defines $UTB^-$.

Additionally, we take a look at the definability relations between axiomatic truth theories and axiomatic theories of definability or skolem functions.

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The use of nonstandard models *N of the natural numbers has recently found several applications in arithmetic Ramsey theory. The basic observation is that every infinite number in *N corresponds to an ultrafilter on N, and the algebra of ultrafilters is a really powerful tool in this field. Note that this notion also makes sense in any model of PA, where one can consider the 1-type of any infinite number.

Furthermore, nonstandard natural numbers are endowed with a natural compact topology, and one can apply the methods of topological dynamics considering the shift operator $x \mapsto x+1$ . This very peculiar dynamic has interesting characteristics.

In this talk I will also present a new result in the style of Hindman’s Theorem about the existence of infinite monochromatic configurations in any finite coloring of the natural numbers. A typical example is the following monochromatic pattern:
a, b, c, $\ldots$ , a+b+ab, a+c+ac, b+c+bc, $\ldots$ , a+b+c+ab+ac+bc+abc.

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In this talk, I will present generalisations of some incompleteness results along two axes: r.e. theories are replaced by $\Sigma_{n+1}$-definable ones, and the base theory is pushed down as far as it will go below PA. Such results are often easy to prove from suitably formulated generalisations of facts used in the original proofs. I will present a handful of such facts, including versions of the arithmetised completeness theorem and the Orey–Hájek characterisation, to show what additional assumptions our theories must satisfy for the results to generalise. Two salient classes of theories emerge in this context: (a) $\Sigma_n$-sound extensions of I$\Sigma_n$ + exp, and (b) $\Pi_n$-complete, consistent extensions of I$\Sigma_{n+1}$. Finally, I will discuss some results that fail to generalise to $\Sigma_{n+1}$-definable theories, as well as an open problem related to Woodin's theorem on the universal algorithm.

The presentation is based on the following paper: https://doi.org/10.1017/S1755020321000307

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This is a joint work with Albert Visser. We prove that no consistent finitely axiomatized theory one-dimensionally interprets its own extension with predicative comprehension. This constitutes a result with the flavor of the Second Incompleteness Theorem whose formulation is completely arithmetic-free. Probably the most important novel feature that distinguishes our result from the previous results of this kind is that it is applicable to arbitrary weak theories, rather than to extensions of some base theory. The methods used in the proof of the main result yield a new perspective on the notion of sequential theory, in the setting of forcing-interpretations. https://arxiv.org/abs/2109.02548

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