Archive for Mathematical Logic最新文献

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.

Realizability notions in mathematical logic have a long history, which can be traced back to the work of Stephen Kleene in the 1940s, aimed at exploring the foundations of intuitionistic logic. Kleene’s initial realizability laid the ground for more sophisticated notions such as Kreisel’s modified realizability and various modern approaches. In this context, our work aligns with the lineage of realizability strategies that emphasize the accumulation, rather than the propagation of precise witnesses. In this paper, we introduce a new notion of realizability, namely herbrandized modified realizability. This novel form of (cumulative) realizability, presented within the framework of semi-intuitionistic logic is based on a recently developed star combinatory calculus, which enables the gathering of witnesses into nonempty finite sets. We also show that the previous analysis can be extended from logic to (Heyting) arithmetic.

We investigate the theory Peano Arithmetic with Indiscernibles ((textrm{PAI})). Models of (textrm{PAI}) are of the form (({mathcal {M}},I)), where ({mathcal {M}}) is a model of (textrm{PA}), I is an unbounded set of order indiscernibles over ({mathcal {M}}), and (({mathcal {M}},I)) satisfies the extended induction scheme for formulae mentioning I. Our main results are Theorems A and B following. Theorem A. Let ({mathcal {M}}) be a nonstandard model of (textrm{PA}) of any cardinality. (mathcal {M }) has an expansion to a model of (textrm{PAI}) iff ( {mathcal {M}}) has an inductive partial satisfaction class. Theorem A yields the following corollary, which provides a new characterization of countable recursively saturated models of (textrm{PA}): Corollary. A countable model ({mathcal {M}}) of (textrm{PA}) is recursively saturated iff ({mathcal {M}}) has an expansion to a model of (textrm{PAI}). Theorem B. There is a sentence (alpha ) in the language obtained by adding a unary predicate I(x) to the language of arithmetic such that given any nonstandard model ({mathcal {M}}) of (textrm{PA}) of any cardinality, ({mathcal {M}}) has an expansion to a model of (text {PAI}+alpha ) iff ({mathcal {M}}) has a inductive full satisfaction class.

A cohesive power of a structure is an effective analog of the classical ultrapower of a structure. We start with a computable structure, and consider its effective power over a cohesive set of natural numbers. A cohesive set is an infinite set of natural numbers that is indecomposable with respect to computably enumerable sets. It plays the role of an ultrafilter, and the elements of a cohesive power are the equivalence classes of certain partial computable functions determined by the cohesive set. Thus, unlike many classical ultrapowers, a cohesive power is a countable structure. In this paper we focus on cohesive powers of graphs, equivalence structures, and computable structures with a single unary function satisfying various properties, which can also be viewed as directed graphs. For these computable structures, we investigate the isomorphism types of their cohesive powers, as well as the properties of cohesive powers when they are not isomorphic to the original structure.

We mainly investigate models of set theory with restricted choice, e.g., ZF + DC + the family of countable subsets of (lambda ) is well ordered for every (lambda ) (really local version for a given (lambda )). We think that in this frame much of pcf theory, (and combinatorial set theory in general) can be generalized. We prove here, in particular, that there is a proper class of regular cardinals, every large enough successor of singular is not measurable and we can prove cardinal inequalities. Solving some open problems, we prove that if (mu> kappa = textrm{cf}(mu ) > aleph _{0},) then from a well ordering of ({mathscr {P}}({mathscr {P}}(kappa )) cup {}^{kappa >} mu ) we can define a well ordering of ({}^{kappa } mu .)