I will continue my overview of Schmerl’s work on generalizations of Tennenbaum’s theorm to general reducts of Peano Arithmetic.

I will continue discussing work of Schmerl addressing the question of under what conditions a given reduct of PA has the property that for any non-standard model M of PA the restriction of M to the reduct must be non-computable.

The study of possible semantics for arithmetized languages in nonstandard models has been a lively research field since the seminal paper of A. Robinson “On languages based on non-standard arithmetic”. Our primary inspiration for examining mathematical features of such structures, and recursively saturated in particular, is that every countable recursively saturated model of Peano Arithmetic supports a great variety of nonstandard satisfaction classes that can serve as models for formal theories of truth – those models allow to investigate the role of arithmetic induction in semantic considerations. On the other hand, nonstandard satisfaction classes are used as a tool in model theoretic constructions providing answers to questions in the model theory of formal arithmetic and often make it possible to solve problems that do not explicitly involve nonstandard semantics. In the series of talks, I will present proofs of classical results concerning satisfaction classes and recursive saturation in models of PA.

I will begin talking about work of Schmerl on generalizations of Tennenbaum’s theorem on the non-exsistence of computable nonstandard models of PA to reducts of PA.

The study of possible semantics for arithmetized languages in nonstandard models has been a lively research field since the seminal paper of A. Robinson “On languages based on non-standard arithmetic”. Our primary inspiration for examining mathematical features of such structures, and recursively saturated in particular, is that every countable recursively saturated model of Peano Arithmetic supports a great variety of nonstandard satisfaction classes that can serve as models for formal theories of truth – those models allow to investigate the role of arithmetic induction in semantic considerations. On the other hand, nonstandard satisfaction classes are used as a tool in model theoretic constructions providing answers to questions in the model theory of formal arithmetic and often make it possible to solve problems that do not explicitly involve nonstandard semantics. In the series of talks, I will present proofs of classical results concerning satisfaction classes and recursive saturation in models of PA.

The study of possible semantics for arithmetized languages in nonstandard models has been a lively research field since the seminal paper of A. Robinson “On languages based on non-standard arithmetic”. Our primary inspiration for examining mathematical features of such structures, and recursively saturated in particular, is that every countable recursively saturated model of Peano Arithmetic supports a great variety of nonstandard satisfaction classes that can serve as models for formal theories of truth – those models allow to investigate the role of arithmetic induction in semantic considerations. On the other hand, nonstandard satisfaction classes are used as a tool in model theoretic constructions providing answers to questions in the model theory of formal arithmetic and often make it possible to solve problems that do not explicitly involve nonstandard semantics. In the series of talks, I will present proofs of classical results concerning satisfaction classes and recursive saturation in models of PA.

In this talk and its prequel I construct models of arithmetic with exactly two expansions to a model of $ACA_0$. Last time, we saw how to build models of arithmetic which are A-rather classless for some class A of the model. In this talk, I will use a kind of forcing argument to show how to pick this A so that the resulting model has exactly two expansions to $ACA_0$. Time permitting, I will explain the difficulties in moving from two to three.

The study of possible semantics for arithmetized languages in nonstandard models has been a lively research field since the seminal paper of A. Robinson “On languages based on non-standard arithmetic”. Our primary inspiration for examining mathematical features of such structures, and recursively saturated in particular, is that every countable recursively saturated model of Peano Arithmetic supports a great variety of nonstandard satisfaction classes that can serve as models for formal theories of truth – those models allow to investigate the role of arithmetic induction in semantic considerations. On the other hand, nonstandard satisfaction classes are used as a tool in model theoretic constructions providing answers to questions in the model theory of formal arithmetic and often make it possible to solve problems that do not explicitly involve nonstandard semantics. In the series of talks, I will present proofs of classical results concerning satisfaction classes and recursive saturation in models of PA.

In this talk and its sequel I will construct models of arithmetic with exactly two expansions to a model of $ACA_0$. To do so, I will use a modified version of Keisler’s construction of a rather-classless model. This talk will focus on this construction, while in Part 2 I will show how to use this construction to get the result.

In this talk, I will discuss the relationship between “strong” models of PA($Q^2$) and $kappa$-like models. Namely, if $kappa$ is a regular uncountable cardinal, then every countable “weak” model has a $kappa$-like ($Q^2$)-elementary end extension that is a kappa like strong model, and if M is a strong model it is $kappa$ like for some $kappa$.

We can extend the language of first order logic to add in a new quantifier, $Q^2$, which binds two free variables. The intended interpretation of $Q^2 x,y\phi(x, y)$ is “There is an infinite (unbounded) set $X$ such that $\phi(x, y)$ holds for each $x \neq y \in X.$” The theory PA($Q^2$) is the theory of Peano Arithmetic in this augmented language (asserting that induction holds for all formulas, including with Ramsey quantifier) and can be thought of as a second order theory whose models are of the form $(M, \mathfrak{X})$ where $\mathfrak{X} \subseteq P(M)$. In this talk, I will present a few results due to Macintyre (1980) and Schmerl & Simpson (1982), namely that models of PA($Q^2$) correspond to models of the second order system $\Pi_1^1-CA_0$. If there is time, I will present Macintyre’s proof that so-called “strong” models of this theory correspond to $\kappa$-like models for some regular $\kappa$.

In previous semesters of this seminar, I have talked about how the technique of forcing, originally developed by Cohen for building models of set theory, can be used to produce models of arithmetic with various properties. In this talk, I will present a forcing proof of Harrington’s theorem on the conservativity of $\mathsf{WKL}_0$ over $\mathsf{RCA}_0$. More formally, any countable model of $\mathsf{RCA}_0$ can be extended to a model of $\mathsf{WKL}_0$ with the same first-order part. As an immediate corollary, we get that any $\Pi^1_1$ sentence provable by $\mathsf{WKL}_0$ is already provable by $\mathsf{RCA}_0$.

I will present a characterization of the closed normal subgroups of the automorphism group of pseudo-recursively saturated models of Presburger arithmetic using the machinery developed in the prior three talks.

I will present a characterization of the closed normal subgroups of the automorphism group of pseudo-recursively saturated models of Presburger arithmetic using the machinery developed in the prior three talks.

This is a continuation of last week’s talk. I will continue with a proof of Woodin’s theorem, recently generalized by Blanck and Enayat, showing that for every computably enumerable theory $T$ extending ${\rm PA}$, there is a corresponding index $e$ such that ${\rm PA}\vdash “W_e$ is finite” and whenever a model $M\models T$ satisfies that $W_e$ is contained in some $M$-finite set $s$, then $M$ has an end-extension $N\models T$ in which $W_e=s$. Indeed, the hypotheses can be relaxed to say that $T$ extends ${\rm I}\Sigma_1$, but I will not discuss this in the talk.

An extended abstract can be found here on my blog.

Jim Schmerl recently proved that there are continuum many theories extending PA, for which whenever M and N are countable arithmetically saturated models of two different such theories, their automorphism groups are non isomorphic. In part I, we give a survey of results concerning automorphism groups of countable arithmetically saturated models of PA, and introduce notions and prove results which will be used in part II to prove Schmerl’s Theorem.

The focus of this talk is the question of what a computable process can output by passing to a nonstandard model of arithmetic. It is not difficult to see that a computable process can change its output by passing to a nonstandard model, but in fact, for some processes, we can thus affect any arbitrary desired change. I will discuss and prove a theorem of Woodin, recently generalized by Blanck and Enayat, showing that for every computably enumerable theory $T$ extending ${\rm PA}$, there is a corresponding index $e$ such that ${\rm PA}\vdash “W_e$ is finite” and whenever a model $M\models T$ satisfies that $W_e$ is contained in some $M$-finite set $s$, then $M$ has an end-extension $N\models T$ in which $W_e=s$. Indeed, the hypotheses can be relaxed to say that $T$ extends ${\rm I}\Sigma_1$, but I will not discuss this in the talk.

An extended abstract can be found here on my blog.

Jim Schmerl recently proved that there are continuum many theories extending PA, for which whenever M and N are countable arithmetically saturated models of two different such theories, their automorphism groups are non isomorphic. In part I, we give a survey of results concerning automorphism groups of countable arithmetically saturated models of PA, and introduce notions and prove results which will be used in part II to prove Schmerl’s Theorem.

I will discuss a recent result of Gabe Conant that there is no structure lying properly between (Z,+) and (Z,+,<).

This talk will focus on some results on the automorphism groups of countable, pseudo-recursively saturated models of Presburger arithmetic and on the characterization of their closed normal subgroups.