NYLogic

One of the most important constructions is model theory is the ultrapower construction. We will discuss a computability-theoretic analogue of an ultrapower where non-principal ultrafilters, which are non-constructive, are replaced by cohesive sets. A cohesive set is a set that cannot be spit by any computably enumerable set. The resultant structure, known as the cohesive power, manages to be computable and constructive while maintaining analogues of the properties of ultrapowers. We will review recent results applying the cohesive power construction to algebraic extensions of Q.

Prikry forcing was devised by Karel Prikry to show that, given large cardinals, one can make a regular cardinal singular while having it remain a cardinal. The cofinal sequence added by Prikry forcing is called the Prikry sequence. The Prikry sequence is maximal in the sense that any other generic sequence is, modulo a finite initial segment, a subsequence of the Prikry sequence. We call this property the maximality property. A key to showing this property is using the normal ultrafilter associated with the singularized large cardinal. In this talk, we discuss the maximality property for Prikry forcings of various ultrafilters. We present partial results on a conjecture of Woodin on maximality for supercompact Prikry forcing, Prikry forcings without the maximality property, and intermediate models of Prikry forcings.

Slides

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Given a cardinal $\kappa$, a set is $\lt\kappa$-blurrily ordinal definable if it belongs to an OD set of cardinality less than $\kappa$, and the $\lt\kappa$-blurry HOD, denoted $\lt\kappa$-HOD, is the collection of all hereditarily $\lt\kappa$-blurrily OD sets. This is a weakly increasing hierarchy of inner models, beginning with HOD and whose union is the whole universe V (assuming choice in V). This hierarchy was introduced by Fuchs, with precursors (the cases $\kappa=\omega,\omega_1$) in the work of Hamkins and Leahy, and Tzouvaras. The leaps are the indices of the hierarchy where a new model occurs, and the possible structure of the leaps has been studied quite a bit by Fuchs, but the question whether the corresponding models satisfy the axiom of choice or not has not been investigated in generality so far. Let’s say that a leap is an AC-leap if the corresponding model in the blurry HOD hierarchy satisfies AC, and otherwise, it is a non-AC-leap. The main theme of this work is to gain a better understanding of the possible AC/non AC patterns in the structure of leaps.

Trivially, it is consistent that every level of the hierarchy satisfies choice, say, in a model of V=HOD (in which case there are no leaps). Meanwhile, it is part of the basic structure theory of leaps due to Fuchs that every limit of leaps is a non-AC-leap. It was observed by Hamkins and Leahy that (in the current terminology) $\lt\omega-HOD=HOD$, so $\omega$ is not a leap. The only published result on successor leaps which are non-AC leaps is due to Kanovei, whereby making use of a product of Jensen forcing a forcing extension of L is obtained in which $\omega_1$ is a non-AC leap. We will show two ways to generalize this construction to larger cardinals. The first obvious idea is to use the generalization of Jensen’s forcing to inaccessible $\kappa$ due to Friedman & Gitman in order to produce forcing extensions of $L$ where $\kappa^+$ is the least leap, and a non-AC leap, and GCH holds. The other generalization is to a cardinal of the form $\kappa^+$ such that $\kappa$ is regular and a certain $\diamondsuit$ assumption holds (which is always true in $L$ in this situation); the forcing is a free Suslin tree, and the argument that this works builds on recent work of Krueger.

Along the way, we will isolate the requisite properties of the forcings involved and arrive at the notions of $\kappa$-Kanovei and $\kappa$-Jensen posets.

This is joint work with my advisor, Gunter Fuchs.

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In 1961 Abraham Robinson solved a centuries-old problem by developing rigorous foundations for infinitesimal calculus. His model-theoretic approach was criticized for its dependence on the axiom of choice and its lack of categoricity. I will argue that the axiomatic approach can overcome these objections.

Starting with the ideas that can be found in the writings of Leibniz and other early infinitesimalists, I will present the theory SPOT, a conservative extension of ZF, that is capable of developing elementary analysis via infinitesimals. Natural generalizations then lead to theories that enable techniques covering almost the entire spectrum of nonstandard analysis. The final theory in the sequence, BST, is 'categorical over ZFC.' Similar results are obtained for theories with multiple levels of standardness. A further extension of the language of these theories allows for a simple presentation of recent results of R. Jin and M. Di Nasso; I will give Jin’s proof of Ramsey’s theorem as an example.

The set-theoretic view of the Leibnizian continuum presents a challenge to traditional set theory, as the existence of infinitesimals entails the existence of unlimited ('infinite') natural numbers. I will indicate how the above theories can be formulated from an 'external' point of view, in terms of an embedding of the standard universe into the internal universe.

This is joint work with Mikhail G. Katz.

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I shall introduce the elementary theory of surreal arithmetic (SA), a first-order theory that is true in the surreal field when equipped with its birthday order structure. This structure, I shall prove, is bi-interpretable with the set-theoretic universe $(V,\in)$, and indeed the theory of surreal arithmetic SA is bi-interpretable with ZFC. This is joint work in progress with myself, Junhong Chen, and Ruizhi Yang, of Fudan University, Shanghai.

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