Narrow Results

This paper answers a question of Usuba [Extendible cardinals and the mantle, Arch. Math. Logic58(1–2) (2019) 71–75], establishing the optimality of the large cardinal assumption of his remarkable theorem that if there is an extendible cardinal, there is a minimum inner model from which the universe of sets can be obtained as a forcing extension.

We investigate higher dimensional chain conditions, where the largeness notion is given by Fubini products of a given ideal. From strong saturation properties of an ideal, we derive abstractly versions of higher dimensional $\mathrm{\Delta }$-system lemma, which imply many posets, including any finite support iteration of $\sigma$-centered posets and measure algebras, satisfy the higher dimensional chain conditions. We then show that if a poset satisfies a strengthening of the $\sigma$-finite chain condition by Horn and Tarski, then it satisfies higher dimensional chain conditions. As an application, we derive Ramsey-theoretic consequences, namely various partition hypotheses as studied by Bannister, Bergfalk, Moore and Todorcevic, from the existence of ideals satisfying strong chain conditions.

What is now called Chaos pertains to generic autonomous behavior of dynamical systems, and is in particular associated with strangeness. From this genericity, conclusion is being drawn that chaos is widespread in nature, particularly in Biology. Hence, chaos provides an alternate modelling paradigm, opposed to the idea of behavior of material systems as a superposition of a simple autonomous dynamics (inherently nongeneric) plus the effects of external, time-dependent forcing terms. We discuss the history and implications of these several points of view, as a clash between two traditions and two different genericities; one in the autonomous dynamics (chaos) and the other in environment.

Since the work of Gödel and Cohen, which showed that Hilbert's First Problem (the Continuum Hypothesis) was independent of the usual assumptions of mathematics (axiomatized by Zermelo–Fraenkel Set Theory with the Axiom of Choice, ZFC), there have been a myriad of independence results in many areas of mathematics.

These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond(♢) and square(□) discovered by Jensen. Simultaneously, attempts have been made to find suitable natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of cardinal arithmetic that is largely determined inside ZFC.

In this paper we consider interactions between these three theories in the context of singular cardinals, focusing on the various implications between square and scales (a fundamental notion in PCF theory), and on consistency results between relatively strong forms of square and stationary set reflection.

We prove that if λ is a strong limit singular cardinal and κ a regular uncountable cardinal < λ, then NS_κλ, the non-stationary ideal over , is nowhere precipitous. We also show that under the same hypothesis every stationary subset of can be partitioned into λ^κ disjoint stationary sets.

We lift Jensen's coding method into the context of Woodin cardinals. By a theorem of Woodin, any real which preserves a "strong witness" to Woodinness is set-generic. We show however that there are class-generic reals which are not set-generic but preserve Woodinness, using "weak witnesses".

Certain separation problems in descriptive set theory correspond to a forcing preservation property, with a fusion type infinite game associated to it. As an application, it is consistent with the axioms of set theory that the circle 𝕋 can be covered by ℵ₁ many closed sets of uniqueness while a much larger number of H-sets is necessary to cover it.

We extend theories of reverse mathematics by a non-principal ultrafilter, and show that these are conservative extensions of the usual theories ACA₀, ATR₀, and .

We develop a general framework for forcing with coherent adequate sets on $H(\lambda )$ as side conditions, where $\lambda \ge {\omega }_2$ is a cardinal of uncountable cofinality. We describe a class of forcing posets which we call coherent adequate type forcings. The main theorem of the paper is that any coherent adequate type forcing preserves CH. We show that there exists a forcing poset for adding a club subset of ${\omega }_2$ with finite conditions while preserving CH, solving a problem of Friedman [Forcing with finite conditions, in Set Theory: Centre de Recerca Matemática, Barcelona, 2003–2004, Trends in Mathematics (Birkhäuser-Verlag, 2006), pp. 285–295.].

We investigate the relationship between weak square principles and simultaneous reflection of stationary sets.

We prove that the null ideal restricted to a non-null set of reals could be isomorphic to a variety of sigma ideals. Using this, we show that the following are consistent: (1) There is a non-null subset of plane each of whose non-null subsets contains three collinear points. (2) There is a partition of a non-null set of reals into null sets, each of size ${\aleph }_1$, such that every transversal of this partition is null.

Rado’s Conjecture is a compactness/reflection principle that says any nonspecial tree of height ${\omega }_1$ has a nonspecial subtree of size ${\aleph }_1$. Though incompatible with Martin’s Axiom, Rado’s Conjecture turns out to have many interesting consequences that are also implied by certain forcing axioms. In this paper, we obtain consistency results concerning Rado’s Conjecture and its Baire version. In particular, we show that a fragment of $\mathrm{\text{PFA}}$, which is the forcing axiom for Baire Indestructibly Proper forcings, is compatible with the Baire Rado’s Conjecture. As a corollary, the Baire Rado’s Conjecture does not imply Rado’s Conjecture. Then we discuss the strength and limitations of the Baire Rado’s Conjecture regarding its interaction with stationary reflection principles and some families of weak square principles. Finally, we investigate the influence of Rado’s Conjecture on some polarized partition relations.

We introduce the large-cardinal notions of $\xi$-greatly-Mahlo and $\xi$-reflection cardinals and prove (1) in the constructible universe, $L$, the first $\xi$-reflection cardinal, for $\xi$ a successor ordinal, is strictly between the first $\xi$-greatly-Mahlo and the first ${\mathrm{\Pi }}_{\xi }^1$-indescribable cardinals, (2) assuming the existence of a $\xi$-reflection cardinal $\kappa$ in $L$, $\xi$ a successor ordinal, there exists a forcing notion in $L$ that preserves cardinals and forces that $\kappa$ is $(\xi +1)$-stationary, which implies that the consistency strength of the existence of a $(\xi +1)$-stationary cardinal is strictly below a ${\mathrm{\Pi }}_{\xi }^1$-indescribable cardinal. These results generalize to all successor ordinals $\xi$ the original same result of Mekler–Shelah [A. Mekler and S. Shelah, The consistency strength of every stationary set reflects, Israel J. Math.67(3) (1989) 353–365] about a $2$-stationary cardinal, i.e. a cardinal that reflects all its stationary sets.

If $\mathrm{\Gamma }$ is a closed Noetherian graph on a $\sigma$-compact Polish space with no infinite cliques, it is consistent with the choiceless set theory ZF$+$DC that $\mathrm{\Gamma }$ is countably chromatic and there is no Vitali set.

We show that it is relatively consistent with ZFC that there is a non-meager set of reals X such that for every non-meager $Y\subseteq X$, there exist distinct $x,y,z\in Y$ such that z is computable from the Turing join of x and y.

In this article, we develop the forcing method on nonstandard models of bounded arithmetic which is initiated by G. Takeuti and M. Yasumoto. We re-formalize their method in two-sort setting and construct generic extensions for subclasses PTIME. We also relate separation problems of complexity classes to properties of generic extensions.

The forcing method is a powerful tool to prove the consistency of settheoretic assertions relative to the consistency of the axioms of set theory. Laver’s theorem and Bukovský’s theorem assert that set-generic extensions of a given ground model constitute a quite reasonable and sufficiently general class of standard models of set-theory.

Beth model is one of the tools in intuitionistic proof theory. Van Dalen constructed a Beth model for intuitionistic analysis with choice sequences. Bernini and Wendel investigated intuitionistic theories with many types of choice sequences, n-functionals, and their relation with classical type theory. In this paper we construct a Beth model for a basic intuitionistic theory with many types of functionals using recursive approach to define nodes in the Beth model. This model is a tool for justifying consistency of intuitionistic principles for functionals of high types. The principles that hold in our Beth model can be added to the initial basic theory in order to develop it into a relatively strong intuitionistic theory, where a significant part of classical type theory can be interpreted, with the purpose of contributing to the programme of justifying classical mathematics from the intuitionistic point of view. In this paper we show that Kripke's schema for n-functionals and standard axioms for lawless n-functionals hold in our Beth model.