Archive for Mathematical Logic最新文献

In the book Cardinal Invariants on Boolean Algebras by J. Donald Monk many such cardinal functions are defined and studied. Among them several are generalizations of well known cardinal characteristics of the continuum. Alongside a long list of open problems is given. Focusing on half a dozen of those cardinal invariants some of those problems are given an answer here, which in most of the cases is a definitive one. Most of them can be divided in two groups. The problems of the first group ask about the change on those cardinal functions when going from a given infinite Boolean algebra to its simple extensions, while in the second group the comparison is between a couple of given infinite Boolean algebras and their free product.

We define and investigate versions of Silver and Mathias forcing with respect to lower and upper density. We focus on properness, Axiom A, chain conditions, preservation of cardinals and adding Cohen reals. We find rough forcings that collapse (2^omega ) to (omega ), while others are surprisingly gentle. We also study connections between regularity properties induced by these parametrized forcing notions and the Baire property.

The following two assertions are equivalent for an o-minimal expansion of an ordered group (mathcal M=(M,<,+,0,ldots )). There exists a definable bijection between a bounded interval and an unbounded interval. Any definable continuous function (f:A rightarrow M) defined on a definable closed subset of (M^n) has a definable continuous extension (F:M^n rightarrow M).

We prove Robinson consistency theorem as well as Craig, Lyndon and Herbrand interpolation theorems in linear continuous logic.

We show that the Axiom of Countable Choice is necessary and sufficient to prove that the existence of a Borel measure on a pseudometric space such that the measure of open balls is positive and finite implies separability of the space. In this way a negative answer to an open problem formulated in Górka (Am Math Mon 128:84–86, 2020) is given. Moreover, we study existence of maximal (delta )-separated sets in metric and pseudometric spaces from the point of view the Axiom of Choice and its weaker forms.

We prove a topological completeness theorem for the modal logic (textsf{GLP}) containing operators ({langle xi rangle :xi in textsf{Ord}}) intended to capture a wellordered sequence of consistency operators increasing in strength. More specifically, we prove that, given a tall-enough scattered space X, any sentence (phi ) consistent with (textsf{GLP}) can be satisfied on a polytopological space based on finitely many Icard topologies constructed over X and corresponding to the finitely many modalities that occur in (phi ).

In the first half of this paper, we study the way that sets of real numbers closed under Turing equivalence sit inside the real line from the perspective of algebra, measure and orders. Afterwards, we combine the results from our study of these sets as orders with a classical construction from Avraham to obtain a restriction about how non trivial automorphism of the Turing degrees (if they exist) interact with 1-generic degrees.

A (sigma )-ideal (mathcal {I}) on a Polish group ((X,+)) has the Smital Property if for every dense set D and a Borel (mathcal {I})-positive set B the algebraic sum (D+B) is a complement of a set from (mathcal {I}). We consider several variants of this property and study their connections with the countable chain condition, maximality and how well they are preserved via Fubini products. In particular we show that there are (mathfrak {c}) many maximal invariant (sigma )-ideals with Borel bases on the Cantor space (2^omega ).