Archive for Mathematical Logic最新文献

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.

It is shown that both classical and intuitionistic propositional logics of an associative binary modality are undecidable. The proof is based on the deduction theorem for these logics.

We investigate quantifier-free induction for Lisp-like lists constructed inductively from the empty list ( nil ) and the operation ({textit{cons}}), that adds an element to the front of a list. First we show that, for (m ge 1), quantifier-free (m)-step induction does not simulate quantifier-free ((m + 1))-step induction. Secondly, we show that for all (m ge 1), quantifier-free (m)-step induction does not prove the right cancellation property of the concatenation operation on lists defined by left-recursion.

We prove that every weakly square compact cardinal is a strong limit cardinal, and therefore weakly compact. We also study Aronszajn trees with no uncountable finitely splitting subtrees, characterizing them in terms of being Lindelöf with respect to a particular topology. We prove that the class of such trees is consistently non-empty and lies between the classes of Suslin and Aronszajn trees.

For a free filter F on (omega ), endow the space (N_F=omega cup {p_F}), where (p_Fnot in omega ), with the topology in which every element of (omega ) is isolated whereas all open neighborhoods of (p_F) are of the form (Acup {p_F}) for (Ain F). Spaces of the form (N_F) constitute the class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Josefson–Nissenzweig theorem from Banach space theory. We prove, e.g., that, for a filter F, the space (N_F) carries a sequence (langle mu _n:nin omega rangle ) of normalized finitely supported signed measures such that (mu _n(f)rightarrow 0) for every bounded continuous real-valued function f on (N_F) if and only if (F^*le _K{mathcal {Z}}), that is, the dual ideal (F^*) is Katětov below the asymptotic density ideal ({mathcal {Z}}). Consequently, we get that if (F^*le _K{mathcal {Z}}), then: (1) if X is a Tychonoff space and (N_F) is homeomorphic to a subspace of X, then the space (C_p^*(X)) of bounded continuous real-valued functions on X contains a complemented copy of the space (c_0) endowed with the pointwise topology, (2) if K is a compact Hausdorff space and (N_F) is homeomorphic to a subspace of K, then the Banach space C(K) of continuous real-valued functions on K is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space K contains a non-trivial convergent sequence, then the space C(K) is not Grothendieck.

We obtain several game characterizations of Baire 1 functions between Polish spaces X, Y which extends the recent result of V. Kiss. Then we propose similar characterizations for equi-Bare 1 families of functions. Also, using related ideas, we give game characterizations of Baire measurable and Lebesgue measurable functions.

We construct a model with a saturated ideal I over ({mathcal {P}}_{kappa }lambda ) and study the extent of saturation of I.