A combinatorial result related to the consistency of New Foundations

We prove a combinatorial result for models of the 4-fragment of the Simple Theory of Types (TST), TST₄. The result says that if A=〈A₀,A₁,A₂,A₃〉 is a standard transitive and rich model of TST₄, then A satisfies the 〈0,0,n〉-property, for all n≥2. This property has arisen in the context of the consistency problem of the theory New Foundations (NF). The result is a weak form of the combinatorial condition (existence of ω-extendible coherent triples) that was shown in Tzouvaras (2007) [5] to be equivalent to the consistency of NF. Such weak versions were introduced in Tzouvaras (2009) [6] in order to relax the intractability of the original condition. The result strengthens one of the main theorems of Tzouvaras (2007) [5, Theorem 3.6] which is just equivalent to the 〈0,0,2〉-property.     

Generic relativizations of fine structure

It is shown how certain generic extensions of a fine structural model in the sense of Mitchell and Steel [MiSt] can be reorganized as relativizations of the model to the generic object. This is then applied to the construction of Steel's core model for one Woodin cardinal [St] and its generalizations. Received: 28 March 1997     

Fusion and large cardinal preservation

In this paper we introduce some fusion properties of forcing notions which guarantee that an iteration with supports of size ?κ   not only does not collapse κ⁺${\kappa }^{+}$ but also preserves the strength of κ (after a suitable preparatory forcing). This provides a general theory covering the known cases of tree iterations which preserve large cardinals (cf. Dobrinen and Friedman (2010) [3], Friedman and Halilovi? (2011) [5], Friedman and Honzik (2008) [6], Friedman and Magidor (2009) [8], Friedman and Zdomskyy (2010) [10], Honzik (2010) [12]).     

On the weak Freese–Nation property of ?(ω)

Continuing [6], [8] and [16], we study the consequences of the weak Freese-Nation property of (?(ω),⊆). Under this assumption, we prove that most of the known cardinal invariants including all of those appearing in Cichoń's diagram take the same value as in the corresponding Cohen model. Using this principle we could also strengthen two results of W. Just about cardinal sequences of superatomic Boolean algebras in a Cohen model. These results show that the weak Freese-Nation property of (?(ω),⊆) captures many of the features of Cohen models and hence may be considered as a principle axiomatizing a good portion of the combinatorics available in Cohen models. Received: 7 June 1999 / Revised version: 17 October 1999 /?Published online: 15 June 2001     

Cardinal characteristics for Menger-bounded subgroups

Machura, Shelah and Tsaban showed in [M. Machura, S. Shelah, B. Tsaban, Squares of Menger-bounded groups, Trans. Amer. Math. Soc., in press, http://arxiv.org/pdf/math.GN/0611353, 2007] that under the condition, that a relative d^′(P) of the dominating number is at least d, there are subgroups of the Baer-Specker group whose kth power is Menger-bounded and whose (k+1)st power is not. We show that the sufficient condition implies r?d and indeed can be replaced by r?d. This result includes an affirmative answer to a question by Tsaban on a possibly weaker still sufficient condition. We show that it is consistent relative to ZFC that g?r<d and there are subgroups of the Baer-Specker group whose kth power is Menger-bounded and whose (k+1)st power is not.     

On successors of Jónsson cardinals

We show that, like singular cardinals, and weakly compact cardinals, Jensen's core model K for measures of order zero [4] calculates correctly the successors of Jónsson cardinals, assuming does not exist. Namely, if is a Jónsson cardinal then , provided that there is no non-trivial elementary embedding . There are a number of related results in ZFC concerning in V and inner models, for a Jónsson or singular cardinal. Received: 8 December 1998     

The differences between Kurepa trees and Jech-Kunen trees

Summary By an ₁ we mean a tree of power ₁ and height ₁. An ₁-tree is called a Kurepa tree if all its levels are countable and it has more than ₁ branches. An ₁-tree is called a Jech-Kunen tree if it has branches for some strictly between ₁ and . In Sect. 1, we construct a model ofCH plus , in which there exists a Kurepa tree with not Jech-Kunen subtrees and there exists a Jech-Kunen tree with no Kurepa subtrees. This improves two results in [Ji1] by not only eliminating the large cardinal assumption for [Ji1, Theorem 2] but also handling two consistency proofs of [Ji1, Theorem 2 and Theorem 3] simultaneously. In Sect. 2, we first prove a lemma saying that anAxiom A focing of size ₁ over Silver's model will not produce a Kurepa tree in the extension, and then we apply this lemma to prove that, in the model constructed for Theorem 2 in [Ji1], there exists a Jech-Kunen tree and there are no Kurepa trees.     

PROPOSITIONAL LOGIC,FRAMES, AND FUZZY ALGEBRA

Abstract

This paper offers a new look at such things as the fuzzy subalgebras and congruences of an algebra, the fuzzy ideals of a ring or a lattice, and similar entities, by exhibiting them as the models, in the chosen frame T of truth values, of naturally corresponding propositional theories. This provides a systematic approach to the study of the partially ordered sets formed by these various entities, and we demonstrate its usefulness by employing it to derive a number of results, some old and some new, concerning these partially ordered sets. In particular, we prove they are complete lattices, algebraic or continuous, depending on whether T is algebraic or continuous, respectively (Proposition 3); they satisfy the same lattice identities for arbitrary T that hold in the case T = 2 (Corollary of Proposition 4); and they are coherent frames for any coherent T whenever this is the case for T = 2 (Proposition 6). In addition we show, generalizing a result by Makamba and Murali [10], that the familiar classical situations where the congruences of an algebra correspond to certain other entities, such as the normal subgroups of a group or the ideals of a ring, extend to the fuzzy case by proving that the corresponding propositional theories are equivalent (Proposition 2). Further, we obtain the result of Gupta and Kantroo [5] that the fuzzy radical ideals of a commutative ring with unit are the meets of fuzzy prime ideals for arbitrary continuous T in place of the unit interval, using basic facts concerning continuous frames (Proposition 7).     

The strength of extensionality II — Weak weak set theories without infinity

By obtaining several new results on Cook-style two-sorted bounded arithmetic, this paper measures the strengths of the axiom of extensionality and of other weak fundamental set-theoretic axioms in the absence of the axiom of infinity, following the author’s previous work [K. Sato, The strength of extensionality I — weak weak set theories with infinity, Annals of Pure and Applied Logic 157 (2009) 234-268] which measures them in the presence. These investigations provide a uniform framework in which three different kinds of reverse mathematics-Friedman-Simpson’s “orthodox” reverse mathematics, Cook’s bounded reverse mathematics and large cardinal theory-can be reformulated within one language so that we can compare them more directly.     

Foundational aspects of singular integrals

We investigate integration of classes of real-valued continuous functions on (0,1]$(0,1]$. Of course difficulties arise if there is a non-L¹$L^1$ element in the class, and the Hadamard finite part integral (p.f.) does not apply. Such singular integrals arise naturally in many contexts including PDEs and singular ODEs.     

POWERSET OPERATOR BASED FOUNDATION FORPOINT-SET LATTICE-THEORETIC (POSLAT) FUZZY SET THEORIES and TOPOLOGIES

Abstract

This paper sets forth in detail point-set lattice-theoretic or poslat foundations of all mathematical and fuzzy set disciplines in which the operations of taking the image and pre-image of (fuzzy) subsets play a fundamental role; such disciplines include algebra, measure and probability theory, and topology. In particular, those aspects of fuzzy sets, hinging around (crisp) powersets of fuzzy subsets and around powerset operators between such powersets lifted from ordinary functions between the underlying base sets, are examined and characterized using point-set and lattice-theoretic methods. The basic goal is to uniquely derive the powerset operators and not simply stipulate them, and in doing this we explicitly distinguish between the “fixed-basis” case (where the underlying lattice of membership values is fixed for the sets in question) and the “variable-basis” case (where the underlying lattice of membership values is allowed to change). Applications to fuzzy sets/logic include: development and justification/characterization of the Zadeh Extension Principle [36], with applications for fuzzy topology and measure theory; characterizations of ground category isomorphisms; rigorous foundation for fuzzy topology in the poslat sense; and characterization of those fuzzy associative memories in the sense of Kosko [18] which are powerset operators. Some results appeared without proof in [31], some with partial proofs in [32], and some in the fixed-basis case in Johnstone [13] and Manes [22].     

On an obstruction for perfect matchings

Steffens [3] introduced a substructure (called below a “compressed set”) which prevents a graph from having a perfect matching, and proved that a countable graph possesses a perfect matching if and only if it does not contain such a substructure. In this paper we study some properties of compressed sets. Dedicated to Paul Erdős on his seventieth birthday     

The eskolemization of universal quantifiers

This paper is a sequel to the papers Baaz and Iemhoff (2006, 2009) [4] and [6] in which an alternative skolemization method called eskolemization was introduced that, when restricted to strong existential quantifiers, is sound and complete for constructive theories. In this paper we extend the method to universal quantifiers and show that for theories satisfying the witness property it is sound and complete for all formulas. We obtain a Herbrand theorem from this, and apply the method to the intuitionistic theory of equality and the intuitionistic theory of monadic predicates.     

The distribution of ITRM-recognizable reals

Infinite Time Register Machines (ITRM's) are a well-established machine model for infinitary computations. Their computational strength relative to oracles is understood, see e.g. ,  and . We consider the notion of recognizability, which was first formulated for Infinite Time Turing Machines in [6] and applied to ITRM's in [3]. A real x is ITRM-recognizable iff there is an ITRM-program P   such that P^y$P^y$ stops with output 1 iff y=x$y=x$, and otherwise stops with output 0. In [3], it is shown that the recognizable reals are not contained in the ITRM-computable reals. Here, we investigate in detail how the ITRM  -recognizable reals are distributed along the canonical well-ordering _<L${<}_L$ of Gödel's constructible hierarchy L  . In particular, we prove that the recognizable reals have gaps in _<L${<}_L$, that there is no universal ITRM in terms of recognizability and consider a relativized notion of recognizability.     

Typed lambda-calculus in classical Zermelo-Frænkel set theory

system of simple types  , which uses the intuitionistic propositional calculus, with the only connective →. It is very important, because the well known Curry-Howard correspondence between proofs and programs was originally discovered with it, and because it enjoys the normalization property: every typed term is strongly normalizable. It was extended to second order intuitionistic logic, in 1970, by J.-Y. Girard [4], under the name of system F, still with the normalization property. More recently, in 1990, the Curry-Howard correspondence was extended to classical logic, following Felleisen and Griffin [6] who discovered that the law of Peirce corresponds to control instructions in functional programming languages. It is interesting to notice that, as early as 1972, Clint and Hoare [1] had made an analogous remark for the law of excluded middle and controlled jump instructions in imperative languages. There are now many type systems which are based on classical logic; among the best known are the system LC of J.-Y. Girard [5] and the λμ-calculus of M. Parigot [11]. We shall use below a system closely related to the latter, called the λ _c -calculus [8, 9]. Both systems use classical second order logic and have the normalization property. In the sequel, we shall extend the λ _c -calculus to the Zermelo-Fr?nkel set theory. The main problem is due to the axiom of extensionality. To overcome this difficulty, we first give the axioms of ZF in a suitable (equivalent) form, which we call ZF _ɛ . Received: 6 September 1999 / Published online: 25 January 2001     

Cardinal characteristics at κ in a small u(κ) model

We provide a model where $\mathfrak{u}(\kappa )<2^{\kappa }$ for a supercompact cardinal κ. [10] provides a sketch of how to obtain such a model by modifying the construction in [6]. We provide here a complete proof using a different modification of [6] and further study the values of other natural generalizations of classical cardinal characteristics in our model. For this purpose we generalize some standard facts that hold in the countable case as well as some classical forcing notions and their properties.