Archive for Mathematical Logic最新文献

We prove characterizations of bounded forcing axioms and their strengthenings in terms of weak genericity. These characterizations resemble those due to Jensen and Woodin, used to characterize the full forcing axioms and their strengthenings. The weak genericity characterizations allow us to quickly deduce various other characterizations in terms of generic (Sigma _1)-absoluteness. We also provide a limiting result on the extent of such generic absoluteness type characterizations of certain strengthenings of bounded forcing axioms.

We use indecomposable ultrafilters to answer some questions from Hayut and Karagila (in Comments Math Univ Carolin 60(2):285–298, 2019). It is shown that the bound on the strength of Usuba (in A note on uniform ultrafilters in choiceless context, arXiv:2401.04871v1) is optimal.

This paper explores undecidability in theories of positive characteristic function fields in the “geometric” language of rings (mathcal {L}_F = {0,1,+,times ,F}), where F is a unary predicate for the subset of nonconstant elements of the field. We are motivated by the (still open) question of the decidability of the existential fragment of the (mathcal {L}_F)-theory of (mathbb {F}_p(t)): a variant on Hilbert’s Tenth Problem for (mathbb {F}_p(t)). If K denotes the function field of a curve, and has as a constant subfield C an algebraic extension of an odd characteristic finite field (not algebraically closed), we prove the (forall ^1exists)-fragment of the (mathcal {L}_F)-theory of K is undecidable. We identify an algebraic condition on elements of K that allows existing machinery of Eisenträger and Shlapentokh (used to conclude undecidability of the existential fragment of the theory of K in the language of rings with some constant symbols for elements of (K setminus C)) to apply to our setting. This work is drawn from the author’s PhD thesis [as reported by Tyrrell (Undecidability in some Field Theories, University of Oxford, Oxford, 2023. https://ora.ox.ac.uk/objects/uuid:3f8d7c47-a54a-4f27-b156-d1116b11b92f)].

In this paper, we investigate the poset (textbf{OF}(X)) of free open filters on a given space X. In particular, we characterize spaces for which (textbf{OF}(X)) is a lattice. For each (nin mathbb {N}) we construct a scattered space X such that (textbf{OF}(X)) is order isomorphic to the n-element chain, which implies the affirmative answer to two questions of Mooney. Assuming CH we construct a scattered space X such that (textbf{OF}(X)) is order isomorphic to ((omega +1,ge )). To prove the latter facts we introduce and investigate a new stratification of ultrafilters which depends on scattered subspaces of (beta (kappa )). Assuming the existence of n measurable cardinals, for every (m_0,ldots ,m_{n}in mathbb {N}) we construct a space X such that (textbf{OF}(X)) is order isomorphic to (prod _{i=0}^nm_i). Also, we show that the existence of a metric space possessing a free (omega _1)-complete closed, (G_delta ), (F_{sigma }) or Borel ultrafilter is equivalent to the existence of a measurable cardinal.

In [11] Sklinos proved that any uncountable free group is not (aleph _1)-homogeneous. This was later generalized by Belegradek in [1] to torsion-free residually finite relatively free groups, leaving open whether the assumption of residual finiteness was necessary. In this paper we use methods arising from the classical analysis of relatively free groups in infinitary logic to answer Belegradek’s question in the negative. Our methods are general and they also apply to varieties with torsion, for example we show that if V contains a finite non-nilpotent group, then any uncountable V-free group is not (aleph _1)-homogeneous.

We obtain a relatively simple criterion for when a forcing has the ({<},delta )-approximation property, generalizing a result of Unger. Afterwards we apply this criterion to construct variants of Mitchell Forcing in order to answer questions posed by Mohammadpour.

A classification is given of all the countable homogeneous ordered bipartite graphs, including those in which just one of the two parts is ordered. In non-trivial cases where the whole structure is ordered, the two parts are ordered like the rationals, and may interact in three essentially different ways. If just one side is ordered, then the structure is the unique one arising as a Fraïssé limit.

We continue a study of the relations between two consequences of the Continuum Hypothesis discovered by Wacław Sierpiński, concerning uniform continuity of continuous functions and uniform convergence of sequences of real-valued functions, defined on subsets of the real line of cardinality continuum.

In this paper we introduce and study WF-algebra, the algebraic structure associated with basic weak subintuitionistic logic WF. We also prove the duality between descriptive NB-Neighborhood frames and WF-algebras. Among other things, we prove the amalgamation property for the class of WF-algebras.