Mathematics

We define a potentialist system of ZF-structures, that is, a collection of possible worlds in the language of ZF connected by a binary accessibility relation, achieving a potentialist account of the full background set-theoretic universe V . The definition involves Berkeley cardinals, the strongest known large cardinal axioms, inconsistent with the Axiom of Choice. In fact, as background theory we assume just ZF. It turns out that the propositional modal assertions which are valid at every world of our system are exactly those in the modal theory S4.2. Moreover, we characterize the worlds satisfying the potentialist maximality principle, and thus the modal theory S5, both for assertions in the language of ZF and for assertions in the full potentialist language.

This is a short introductory course to Set Theory, based on axioms of von Neumann--Bernays--Gödel (briefly NBG). The text can be used as a base for a lecture course in Foundations of Mathematics, and contains a reasonable minimum which a good (post-graduate) student in Mathematics should know about foundations of this science.

In this paper we classify all ℵ 0 -categorical and C -minimal sets up to elementary equivalence.

We consider a global semianalytic set defined by real analytic functions definable in an o-minimal structure. When the o-minimal structure is polynomially bounded, we show that the closure of this set is a global semianalytic set defined by definable real analytic functions. We also demonstrate that a connected component of a planar global semianalytic set defined by real analytic functions definable in a substructure of the restricted analytic field is a global semianalytic set defined by definable real analytic functions.

We study closure properties of measurable ultrapowers with respect to Hamkin's notion of "freshness" and show that the extent of these properties highly depends on the combinatorial properties of the underlying model of set theory. In one direction, a result of Sakai shows that, by collapsing a strongly compact cardinal to become the double successor of a measurable cardinal, it is possible to obtain a model of set theory in which such ultrapowers possess the strongest possible closure properties. In the other direction, we use various square principles to show that measurable ultrapowers of canonical inner models only possess the minimal amount of closure properties. In addition, the techniques developed in the proofs of these results also allow us to derive statements about the consistency strength of the existence of measurable ultrapowers with non-minimal closure properties.

We prove that it is consistent that Club Stationary Reflection and the Special Aronszajn Tree Property simultaneously hold on ? 2 , thereby contributing to the study of the tension between compactness and incompactness in set theory. The poset which produces the final model follows the collapse of a weakly compact cardinal first with an iteration of club adding (with anticipation) and second with an iteration specializing Aronszajn trees. In the first part of the paper, we prove a general theorem about specializing Aronszajn trees after forcing with what we call F WC -Strongly Proper posets. This type of poset, of which the Levy collapse is a degenerate example, uses systems of exact residue functions to create many strongly generic conditions. We prove a new result about stationary set preservation by quotients of this kind of poset; as a corollary, we show that the original Laver-Shelah model satisfies a strong stationary reflection principle, though it fails to satisfy the full Club Stationary Reflection. In the second part, we show that the composition of collapsing and club adding (with anticipation) is an F WC -Strongly Proper poset. After proving a new result about Aronszajn tree preservation, we show how to obtain the final model.

In this paper, we propose a generalization of Continuous Logic ([BBHU08]) where the distances take values in suitable co-quantales (in the way as it was proposed in [Fla97]). By assuming suitable conditions (e.g., being co-divisible, co-Girard and a V-domain), we provide, as test questions, a proof of a version of the Tarski-Vaught test (Proposition 3.35) and Los Theorem (Theorem 3.62) in our setting.

We describe the endofunctor H on the category CABA of complete and atomic boolean algebras and complete boolean homomorphisms such that the category Alg(H) of algebras for H is dually equivalent to the category Coalg(P) of coalgebras for the powerset endofunctor P on Set . As a consequence, we derive Thomason duality from Tarski duality.

Given two combinatorial notions \mathsf{P}_0 and \mathsf{P}_1 , can we encode \mathsf{P}_0 via \mathsf{P}_1 . In this talk we address the question where \mathsf{P}_0 is 3-coloring of integers and \mathsf{P}_1 is product of finitely many 2-colorings of integers. We firstly reduce the question to a lemma which asserts that certain \Pi^0_1 class of colorings admit two members violating a particular combinatorial constraint. Then we took a digression to see how complex does the class has to be so as to maintain the cross constraint. We weaken the two members in the lemma in certain way to address an open question of Cholak, Dzhafarov, Hirschfeldt and Patey, concerning a sort of Weihrauch degree of stable Ramsey's theorem for pairs. It turns out the resulted strengthen of the lemma is a basis theorem for \Pi^0_1 class with additional constraint. We look at several such variants of basis theorem, among them some are unknown. We end up by introducing some results and questions concerning product of infinitely many colorings.

We show that the set of the ground-model reals has strong measure zero (is strongly meager) after adding a single Cohen real (random real). As consequence we prove that the set of the ground-model reals has strong measure zero after adding a single Hechler real.