Archive for Mathematical Logic最新文献

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 ).

We show that Hechler’s forcings for adding a tower and for adding a mad family can be represented as finite support iterations of Mathias forcings with respect to filters and that these filters are ({mathcal {B}})-Canjar for any countably directed unbounded family ({mathcal {B}}) of the ground model. In particular, they preserve the unboundedness of any unbounded scale of the ground model. Moreover, we show that ({mathfrak {b}}=omega _1) in every extension by the above forcing notions.

It is introduced a new algebra ((A, otimes , oplus , *, rightharpoonup , 0, 1)) called (L_PG)-algebra if ((A, otimes , oplus , *, 0, 1)) is (L_P)-algebra (i.e. an algebra from the variety generated by perfect MV-algebras) and ((A,rightharpoonup , 0, 1)) is a Gödel algebra (i.e. Heyting algebra satisfying the identity ((x rightharpoonup y ) vee (y rightharpoonup x ) =1)). The lattice of congruences of an (L_PG) -algebra ((A, otimes , oplus , *, rightharpoonup , 0, 1)) is isomorphic to the lattice of Skolem filters (i.e. special type of MV-filters) of the MV-algebra ((A, otimes , oplus , *, 0, 1)). The variety (mathbf {L_PG}) of (L_PG) -algebras is generated by the algebras ((C, otimes , oplus , *, rightharpoonup , 0, 1)) where ((C, otimes , oplus , *, 0, 1)) is Chang MV-algebra. Any (L_PG) -algebra is bi-Heyting algebra. The set of theorems of the logic (L_PG) is recursively enumerable. Moreover, we describe finitely generated free (L_PG)-algebras.