Existence of optimal ultrafilters and the fundamental complexity of simple theories

In the first edition of Classification Theory, the second author characterized the stable theories in terms of saturation of ultrapowers. Prior to this theorem, stability had already been defined in terms of counting types, and the unstable formula theorem was known. A contribution of the ultrapower characterization was that it involved sorting out the global theory, and introducing nonforking, seminal for the development of stability theory. Prior to the present paper, there had been no such characterization of an unstable class. In the present paper, we first establish the existence of so-called optimal ultrafilters on Boolean algebras, which are to simple theories as Keisler's good ultrafilters are to all theories. Then, assuming a supercompact cardinal, we characterize the simple theories in terms of saturation of ultrapowers. To do so, we lay the groundwork for analyzing the global structure of simple theories, in ZFC, via complexity of certain amalgamation patterns. This brings into focus a fundamental complexity in simple unstable theories having no real analogue in stability.Comment: The revisions aim to separate the set theoretic and model theoretic aspects of the paper to make it accessible to readers interested primarily in one side. We thank the anonymous referee for many thoughtful comment

Joint Laver diamonds and grounded forcing axioms

I explore two separate topics: the concept of jointness for set-theoretic guessing principles, and the notion of grounded forcing axioms. A family of guessing sequences is said to be joint if all of its members can guess any given family of targets independently and simultaneously. I primarily investigate jointness in the case of various kinds of Laver diamonds. In the case of measurable cardinals I show that, while the assertions that there are joint families of Laver diamonds of a given length get strictly stronger with increasing length, they are all equiconsistent. This is contrasted with the case of partially strong cardinals, where we can derive additional consistency strength, and ordinary diamond sequences, where large joint families exist whenever even one diamond sequence does. Grounded forcing axioms modify the usual forcing axioms by restricting the posets considered to a suitable ground model. I focus on the grounded Martin's axiom which states that Martin's axioms holds for posets coming from some ccc ground model. I examine the new axiom's effects on the cardinal characteristics of the continuum and show that it is quite a bit more robust under mild forcing than Martin's axiom itself.Comment: This is my PhD dissertatio

Indeterminateness and `The' Universe of Sets: Multiversism, Potentialism, and Pluralism

In this article, I survey some philosophical attitudes to talk concerning `the' universe of sets. I separate out four different strands of the debate, namely: (i) Universism, (ii) Multiversism, (iii) Potentialism, and (iv) Pluralism. I discuss standard arguments and counterarguments concerning the positions and some of the natural mathematical programmes that are suggested by the various views

Joint Laver Diamonds and Grounded Forcing Axioms

In chapter 1 a notion of independence for diamonds and Laver diamonds is investigated. A sequence of Laver diamonds for κ is joint if for any sequence of targets there is a single elementary embedding j with critical point κ such that each Laver diamond guesses its respective target via j. In the case of measurable cardinals (with similar results holding for (partially) supercompact cardinals) I show that a single Laver diamond for κ yields a joint sequence of length κ, and I give strict separation results for all larger lengths of joint sequences. Even though the principles get strictly stronger in terms of direct implication, I show that they are all equiconsistent. This is contrasted with the case of θ-strong cardinals where, for certain θ, the existence of even the shortest joint Laver sequences carries nontrivial consistency strength. I also formulate a notion of jointness for ordinary ◊κ-sequences on any regular cardinal κ. The main result concerning these shows that there is no separation according to length and a single ◊κ-sequence yields joint families of all possible lengths. In chapter 2 the notion of a grounded forcing axiom is introduced and explored in the case of Martin\u27s axiom. This grounded Martin\u27s axiom, a weakening of the usual axiom, states that the universe is a ccc forcing extension of some inner model and the restriction of Martin\u27s axiom to the posets coming from that ground model holds. I place the new axiom in the hierarchy of fragments of Martin\u27s axiom and examine its effects on the cardinal characteristics of the continuum. I also show that the grounded version is quite a bit more robust under mild forcing than Martin\u27s axiom itself

Some Applications of Set Theory to Model Theory

We investigate set-theoretic dividing lines in model theory. In particular, we are interested in Keisler's order and Borel complexity. Keisler's order is a pre-order on complete countable theories $T$, measuring the saturation of ultrapowers of models of $T$. In Chapter~\ref{SurveyChapter}, we present a self-contained survey on Keisler's order. In Chapter~\ref{KeislerNew}, we uniformize and sharpen several ultrafilter constructions of Malliaris and Shelah. We also investigate the model-theoretic properties detected by Keisler's order among the simple unstable theories. Borel complexity is a pre-order on sentences of $\mathcal{L}_{\omega_1 \omega}$ measuring the complexity of countable models. In Chapter~\ref{ChapterURL}, we describe joint work with Richard Rast and Chris Laskowski on this order. In particular, we connect the Borel complexity of $\Phi \in \mathcal{L}_{\omega_1 \omega}$ with the number of potential canonical Scott sentences of $\Phi$. In Chapter~\ref{ChapterSB}, we introduce the notion of thickness; when $\Phi$ has class-many potential canonical Scott sentences, thickness is a measure of how quickly this class grows in size. In Chapter~\ref{ChapterTFAG}, we describe joint work with Saharon Shelah on the Borel complexity of torsion-free abelian groups

Generalized descriptive set theory and classification theory

Descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper we study the generalization where countable is replaced by uncountable. We explore properties of generalized Baire and Cantor spaces, equivalence relations and their Borel reducibility. The study shows that the descriptive set theory looks very different in this generalized setting compared to the classical, countable case. We also draw the connection between the stability theoretic complexity of first-order theories and the descriptive set theoretic complexity of their isomorphism relations. Our results suggest that Borel reducibility on uncountable structures is a model theoretically natural way to compare the complexity of isomorphism relations

A predicative variant of a realizability tripos for the Minimalist Foundation.

open2noHere we present a predicative variant of a realizability tripos validating the intensional level of the Minimalist Foundation extended with Formal Church thesis.the file attached contains the whole number of the journal including the mentioned pubblicationopenMaietti, Maria Emilia; Maschio, SamueleMaietti, MARIA EMILIA; Maschio, Samuel

Generalized Descriptive Set Theory and Classification Theory

The field of descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper we study the generalization where countable is replaced by uncountable. We explore properties of generalized Baire and Cantor spaces, equivalence relations and their Borel reducibility. The study shows that the descriptive set theory looks very dierent in this generalized setting compared to the classical, countable case. We also draw the connection between the stability theoretic complexity of first-order theories and the descriptive set theoretic complexity of their isomorphism relations. Our results suggest that Borel reducibility on uncountable structures is a model theoretically natural way to compare the complexity of isomorphism relations.Peer reviewe