Archive for Mathematical Logic最新文献

We give several new equivalences of NIP for formulas and new proofs of known results using Talagrand (Ann Probab 15:837–870, 1987) and Haydon et al. (in: Functional Analysis Proceedings, The University of Texas at Austin 1987–1989, Lecture Notes in Mathematics, Springer, New York, 1991). We emphasize that Keisler measures are more complicated than types (even in the NIP context), in an analytic sense. Among other things, we show that for a first order theory T and a formula (phi (x,y)), the following are equivalent:

1. (i)

   (phi ) has NIP with respect to T.

2. (ii)

   For any global (phi )-type p(x) and any model M, if p is finitely satisfiable in M, then p is generalized DBSC definable over M. In particular, if M is countable, then p is DBSC definable over M. (Cf. Definition 3.7, Fact 3.8.)

3. (iii)

   For any global Keisler (phi )-measure (mu (x)) and any model M, if (mu ) is finitely satisfiable in M, then (mu ) is generalized Baire-1/2 definable over M. In particular, if M is countable, (mu ) is Baire-1/2 definable over M. (Cf. Definition 3.9.)

4. (iv)

   For any model M and any Keisler (phi )-measure (mu (x)) over M,

   $$begin{aligned} sup _{bin M}Big |frac{1}{k}sum _{i=1}^kphi (p_i,b)-mu (phi (x,b))Big |rightarrow 0, end{aligned}$$

   for almost every ((p_i)in S_{phi }(M)^{mathbb N}) with the product measure (mu ^{mathbb N}). (Cf. Theorem 4.4.)

5. (v)

   Suppose moreover that T is countable and NIP, then for any countable model M, the space of global M-finitely satisfied types/measures is a Rosenthal compactum. (Cf. Theorem 5.1.)

In set theory without the Axiom of Choice we prove that the assertion “For every metric space (X, d) with a Borel measure (mu ) such that the measure of every open ball is positive and finite, (X, d) is separable.’ is implied by the axiom of choice for countable collections of sets and implies the axiom of choice for countable collections of finite sets. We also show that neither implication is reversible in Zermelo–Fraenkel set theory weakend to permit the existence of atoms and that the second implication is not reversible in Zermelo–Fraenkel set theory. This gives an answer to a question of Dybowski and Górka (Arch Math Logic 62:735–749, 2023. https://doi.org/10.1007/s00153-023-00868-4).

In this paper we consider some fragments of (textsf{IOpen}) (Robinson arithmetic (mathsf Q) with induction for quantifier-free formulas) proposed by Harvey Friedman and answer some questions he asked about these theories. We prove that (mathsf {I(lit)}) is equivalent to (textsf{IOpen}) and is not finitely axiomatizable over (mathsf Q), establish some inclusion relations between (mathsf {I(=)}, mathsf {I(ne )}, mathsf {I(leqslant )}) and (textsf{I} (nleqslant )). We also prove that the set of diophantine equations solvable in models of (mathsf I (=)) is (algorithmically) decidable.

We address some phenomena about the interaction between lower semicontinuous submeasures on ({mathbb {N}}) and (F_{sigma }) ideals. We analyze the pathology degree of a submeasure and present a method to construct pathological (F_{sigma }) ideals. We give a partial answers to the question of whether every nonpathological tall (F_{sigma }) ideal is Katětov above the random ideal or at least has a Borel selector. Finally, we show a representation of nonpathological (F_{sigma }) ideals using sequences in Banach spaces.

We generalise various theorems for finding indiscernible trees and arrays to positive logic: based on an existing modelling theorem for s-trees, we prove modelling theorems for str-trees, str(_0)-trees (the reduct of str-trees that forgets the length comparison relation) and arrays. In doing so, we prove stronger versions for basing—rather than locally basing or EM-basing—str-trees on s-trees and str(_0)-trees on str-trees. As an application we show that a thick positive theory has k-(mathsf {TP_2}) iff it has 2-(mathsf {TP_2})

In this paper we study the set of MV-algebras with given prime spectrum and we introduce the class of spectral MV-algebras. An MV-algebra is spectral if it is generated by the union of all its prime ideals (or proper ideals, or principal ideals, or maximal ideals). Among spectral MV-algebras, special attention is devoted to bipartite MV-algebras. An MV-algebra is bipartite if it admits an homomorphism onto the MV-algebra of two elements. We prove that both bipartite MV-algebras and spectral MV-algebras can be finitely axiomatized in first order logic. We also prove that there is only, up to isomorphism, a set of MV-algebras with given prime spectrum. A further part of the paper is devoted to some relations between bipartite MV-algebras and their states. Recall that a state on an MV-algebra is a generalization of a probability measure on a Boolean algebra. Particular states are the states with Bayes’ property. We show that an MV-algebra admits a state with the Bayes’ property if and only if it is bipartite.

We study 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.

A family (mathcal {I}) of subsets of a set X is an ideal on X if it is closed under taking subsets and finite unions of its elements. An ideal (mathcal {I}) on X is below an ideal (mathcal {J}) on Y in the Katětov order if there is a function (f{: }Yrightarrow X) such that (f^{-1}[A]in mathcal {J}) for every (Ain mathcal {I}). We show that the Hindman ideal, the Ramsey ideal and the summable ideal are pairwise incomparable in the Katětov order, where

• The Ramsey ideal consists of those sets of pairs of natural numbers which do not contain a set of all pairs of any infinite set (equivalently do not contain, in a sense, any infinite complete subgraph),

• The Hindman ideal consists of those sets of natural numbers which do not contain any infinite set together with all finite sums of its members (equivalently do not contain IP-sets that are considered in Ergodic Ramsey theory),

• The summable ideal consists of those sets of natural numbers such that the series of the reciprocals of its members is convergent.