The theory ZFC-, consisting of the usual axioms of ZFC but with the power set axiom removed — specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well-ordered — is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context.

For example, there are models of ZFC- in which a countable union of countable sets is not countable. There are models of ZFC- for which the Los ultrapower theorem fails, even for wellfounded ultrapowers on a measurable cardinal. Moreover, the theory ZFC- is not sufficient to establish that the union of Σ_n and Π_n sets is closed under bounded quantification. Lastly, there are models of ZFC- for which the Gaifman theorem fails, in that there exists cofinal embeddings j:M–>N between ZFC- models that are Σ₁-elementary, but not fully elementary.

Nevertheless, these deficits of ZFC- are completely repaired by strengthening it to the theory obtained by using collection rather than replacement in the axiomatization above. This is joint work with Joel David Hamkins and Victoria Gitman, and it extends prior work of Andrzej Zarach.

arxiv preprint | post at jdh.hamkins.org | post on Victoria Gitman’s blog

We discuss two results relating ideas in Reverse Mathematics to the properties of models of first order arithmetic. The first shows that we can extend second order arithmetic by the existence of a non-principal ultrafilter — a third order property — while remaining conservative. The second result shows that we can extend models of RCA so that any particular set is definable; this allows us to recover some properties of models of Peano arithmetic for models of RCA.

Definable categories are classes of modules closed under direct product, direct limit, and pure submodule. These play an important role in the theory of modules as they are in bijection with the closed sets of the Ziegler spectrum (and thus with the product closed complete theories of modules). Mittag-Leffler objects for such categories (some sort of generalized atomic module) will be introduced and a necessary and sufficient condition in terms of generators and generalized relations will be given for them to exist.

We (with D’Aquino and Kuhlmann) give a valuation theoretic characterization for a real closed field to be recursively saturated. Previously, Kuhlmann, Kuhlmann, Marshall, and Zekavat gave such a characterization for kappa-saturation, for all infinite cardinals kappa. Our result extends the characterization for a divisible ordered abelian group to be recursively saturated found in some unpublished work of Harnik and Ressayre.

Bukowski-Namba forcing preserves aleph_1 and changes the cofinality of  aleph_2 to omega. We lift this to cardinals kappa > aleph_1 : Assuming a measurable cardinal lambda we construct models over which there is a further “Namba-like” forcing which preserves all cardinals <= kappa and changes the cofinality of kappa^+ to omega. Cofinalities different from omega can also be achieved by starting from measurable cardinals of sufficiently strong Mitchell order. Using core model theory one can show that the respective measurable cardinals are also necessary. This is joint work with Dominik Adolf (Münster).

Slides

Recently Montalban and Shore derived precise limits to the amount of determinacy provable in second order arithmetic.  We review some of the results in this area and recent work on lifting this to a setting of ZF^- with a single measurable cardinal.

Slides

This talk surveys work on the development of model theory for classes of finite structures due primarily to Macpherson and the speaker, and to several of Macpherson’s students. The underlying theme of this work has been to bring to the model-theoretic study of classes of finite structures aspects of the model theory of infinite structures.

It is important in the foundations of mathematics that the natural number system is characterizable as a system of 0 and a successor function by second-order logic. In other words, the following Dedekind’s second-order categoricity theorem holds: every Peano system $(P,e,F)$ is isomorphic to the natural number system $(N,0,S)$. In this talk, I will investigate Dedekind’s theorem and other similar statements. We will first do reverse mathematics over $RCA_0$, and then weaken the base theory. This is a joint work with Stephen G. Simpson.

In a nonstandard model of arithmetic, initial segments with no maximum elements are traditionally called cuts. It is known that even if we restrict our attention to cuts that are closed under a fixed family of functions (e.g., multiplication, the primitive recursive functions, or the Skolem functions), the properties of cuts can still vary greatly. I will talk about what genericity means amongst such great variety. This notion of genericity comes from a version of model theoretic forcing devised by Richard Kaye in his 2008 paper. Some ideas were already implicit in the work by Laurence Kirby and Jeff Paris on indicators in the 1970s.

In recent years there has been renewed interest in theories without the independence property (NIP theories), a class of theories including all stable as well as all o-minimal theories.  In this talk we concentrate on theories, T, which expand that of divisible ordered Abelian groups (a natural situation to consider if one is motivated by the study of o-minimal theories) and consider the problem determining the consequences of assuming that T is NIP on the structure of definable sets in models of T.  One quickly realizes that given the great generality of the NIP assumption  in order to address this type of question one wants to consider much stronger variants of not having the independence property.  Thus we are led to the study of definable sets in models of  theories T expanding that of divisible ordered Abelian groups satisfying various very strong forms of the NIP condition such as finite dp-rank, convex orderability, and VC minimality.  In this talk I will survey results in this area and discuss many open problems.

This talk will be based on my paper, A multiverse perspective on the axiom of constructibility.

Set-theorists often argue against the axiom of constructibility V=L on the grounds that it is restrictive, that we have no reason to suppose that every set should be constructible and that it places an artificial limitation on set-theoretic possibility to suppose that every set is constructible.  Penelope Maddy, in her work on naturalism in mathematics, sought to explain this perspective by means of the MAXIMIZE principle, and further to give substance to the concept of what it means for a theory to be restrictive, as a purely formal property of the theory.

In this talk, I shall criticize Maddy’s specific proposal.  For example, it turns out that the fairly-interpreted-in relation on theories is not transitive, and similarly the maximizes-over and strongly-maximizes-over relations are not transitive.  Further, the theory ZFC + `there is a proper class of inaccessible cardinals’ is formally restrictive on Maddy’s proposal, although this is not what she had desired.

Ultimately, I argue that the $Vneq L$ via maximize position loses its force on a multiverse conception of set theory, in light of the classical facts that models of set theory can generally be extended to (taller) models of V=L.  In particular, every countable model of set theory is a transitive set inside a model of V=L.  I shall conclude the talk by explaining various senses in which V=L remains compatible with strength in set theory.

All countable recursively saturated models of Peano Arithmetic have nonstandard satisfaction classes. In fact, each such model has a great variety of nonstandard satisfaction classes. I will survey model theoretic techniques that can be applied to construct many different inductive satisfaction classes, and I will show how, in return, inductive satisfaction classes are used to prove important result about recursively saturated models of PA. I will also pose an open problem concerning a possible converse to Tarski’s undefinability of truth theorem.

Pdf slides for the talk are here: Satisfaction Classes

An abstract of this talk will be added.

The modal logic of forcing arises when one considers a model of set theory in the context of all its forcing extensions, with “true in all forcing extensions” and“true in some forcing extension” as the accompanying modal operators. In this modal language one may easily express sweeping general forcing principles, such as the assertion that every possibly necessary statement is necessarily possible, which is valid for forcing, or the assertion that every possibly necessary statement is true, which is the maximality principle, a forcing axiom independent of but equiconsistent with ZFC. Similarly, the dual modal logic of grounds concerns the modalities “true in all ground models” and “true in some ground model”. In this talk, I shall survey the recent progress on the modal logic of forcing and the modal logic of grounds. This is joint work with Benedikt Loewe and George Leibman.

This will be a talk on some extremely new work. The proof uses finitary digraph combinatorics, including the countable random digraph and higher analogues involving uncountable Fraisse limits, the surreal numbers and the hypnagogic digraph.

The story begins with Ressayre’s remarkable 1983 result that if $M$ is any nonstandard model of PA, with $langletext{HF}^M,{in^M}rangle$ the corresponding nonstandard hereditary finite sets of $M$, then for any consistent computably axiomatized theory $T$ in the language of set theory, with $Tsupset ZF$, there is a submodel $Nsubsetlangletext{HF}^M,{in^M}rangle$ such that $Nmodels T$. In particular, one may find models of ZFC or even ZFC + large cardinals as submodels of $text{HF}^M$, a land where everything is thought to be finite. Incredible! Ressayre’s proof uses partial saturation and resplendency to prove that one can find the submodel of the desired theory $T$.

My new theorem strengthens Ressayre’s theorem, while simplifying the proof, by removing the theory $T$. We need not assume $T$ is computable, and we don’t just get one model of $T$, but rather all models—the fact is that the nonstandard models of set theory are universal for all countable acyclic binary relations. So every model of set theory is a submodel of $langletext{HF}^M,{in^M}rangle$.

Theorem.(JDH) Every countable model of set theory is isomorphic to a submodel of any nonstandard model of finite set theory. Indeed, every nonstandard model of finite set theory is universal for all countable acyclic binary relations.

The proof involves the construction of what I call the countable random $mathbb{Q}$-graded digraph, a countable homogeneous acyclic digraph that is universal for all countable acyclic digraphs, and proving that it is realized as a submodel of the nonstandard model $langle M,in^Mrangle$. Having then realized a universal object as a submodel, it follows that every countable structure with an acyclic binary relation, including every countable model of ZFC, is realized as a submodel of $M$.

Theorem.(JDH) Every countable model $langle M,in^Mrangle$ of ZFC, including the transitive models, is isomorphic to a submodel of its own constructible universe $langle L^M,in^Mrangle$. In other words, there is an embedding $j:Mto L^M$ that is quantifier-free-elementary.

The proof is guided by the idea of finding a universal submodel inside $L^M$. The embedding $j$ is constructed completely externally to $M$.

Corollary.(JDH) The countable models of ZFC are linearly ordered and even well-ordered, up to isomorphism, by the submodel relation. Namely, any two countable models of ZFC with the same well-founded height are bi-embeddable as submodels of each other, and all models embed into any nonstandard model.

The work opens up numerous questions on the extent to which we may expect in ZFC that $V$ might be isomorphic to a subclass of $L$. To what extent can we expect to have or to refute embeddings $j:Vto L$, elementary for quantifier-free assertions?