A theory of sets with the negation of the axiom of infinity

In this paper we introduce a theory of finite sets FST with a strong negation of the axiom of infinity asserting that every set is provably bijective with a natural number. We study in detail the role of the axioms of Power Set, Choice, Regularity in FST, pointing out the relative dependences or independences among them. FST is shown to be provably equivalent to a fragment of Alternative Set Theory. Furthermore, the introduction of FST is motivated in view of a non-standard development. MSC: 03E30, 03E35.     

Smoke and mirrors: Combinatorial properties of small cardinals equiconsistent with huge cardinals

Since the work of Godel and Cohen many questions in infinite combinatorics have been shown to be independent of the usual axioms for mathematics, Zermelo Frankel Set Theory with the Axiom of Choice (ZFC). Attempts to strengthen the axioms to settle these problems have converged on a system of principles collectively known as Large Cardinal Axioms.These principles are linearly ordered in terms of consistency strength. As far as is currently known, all natural independent combinatorial statements are equiconsistent with some large cardinal axiom. The standard techniques for showing this use forcing in one direction and inner model theory in the other direction.The conspicuous open problems that remain are suspected to involve combinatorial principles much stronger than the large cardinals for which there is a current fine-structural inner model theory for.The main results in this paper show that many standard constructions give objects with combinatorial properties that are, in turn, strong enough to show the existence of models with large cardinals are larger than any cardinal for which there is a standard inner model theory.     

Forcing under Anti‐Foundation Axiom: An expression of the stalks

We introduce a new simple way of defining the forcing method that works well in the usual setting under FA, the Foundation Axiom, and moreover works even under Aczel's AFA, the Anti‐Foundation Axiom. This new way allows us to have an intuition about what happens in defining the forcing relation. The main tool is H. Friedman's method of defining the extensional membership relation ∈ by means of the intensional membership relation ε . Analogously to the usual forcing and the usual generic extension for FA‐models, we can justify the existence of generic filters and can obtain the Forcing Theorem and the Minimal Model Theorem with some modifications. These results are on the line of works to investigate whether model theory for AFA‐set theory can be developed in a similar way to that for FA‐set theory. Aczel pointed out that the quotient of transition systems by the largest bisimulation and transition relations have the essentially same theory as the set theory with AFA. Therefore, we could hope that, by using our new method, some open problems about transition systems turn out to be consistent or independent. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Deductive Strength of Various Distributivity Axioms for Boolean Algebras in Set Theory

In this article, we shall show the generalized notions of distributivity of Boolean algebras have essential relations with several axioms and properties of set theory, say the Axiom of Choice, the Axiom of Dependence Choice, the Prime Ideal Theorems, Martin's axioms, Lebesgue measurability and so on.     

Linearization, Dold-Puppe stabilization, and Mac Lane's -construction

In this paper we study linear functors, i.e., functors of chain complexes of modules which preserve direct sums up to quasi-isomorphism, in order to lay the foundation for a further study of the Goodwillie calculus in this setting. We compare the methods of Dold and Puppe, Mac Lane, and Goodwillie for producing linear approximations to functors, and establish conditions under which these methods are equivalent. In addition, we classify linear functors in terms of modules over an explicit differential graded algebra. Several classical results involving Dold-Puppe stabilization and Mac Lane's -construction are extended or given new proofs.

     

Constructive Order Theory

We introduce the notion of constructive suprema and of constructively directed sets. The Axiom of Choice turns out to be equivalent to the postulate that every supremum is constructive, but also to the hypothesis that every directed set admits a function assigning to each finite subset an upper bound. The Axiom of Multiple Choice (which is known to be weaker than the full Axiom of Choice in set theory without foundation) implies a simple set‐theoretical induction principle (SIP), stating that any system of sets that is closed under unions of well‐ordered subsystems and contains all finite subsets of a given set must also contain that set itself. This is not provable without choice principles but equivalent to the statement that the existence of joins for constructively directed subsets of a poset follows from the existence of joins for nonempty well‐ordered subsets. Moreover, we establish the equivalence of SIP with several other fundamental statements concerning inductivity, compactness, algebraic closure systems, and the exchange between chains and directed sets.     

Realisability for infinitary intuitionistic set theory

We introduce a realisability semantics for infinitary intuitionistic set theory that is based on Ordinal Turing Machines (OTMs). We show that our notion of OTM-realisability is sound with respect to certain systems of infinitary intuitionistic logic, and that all axioms of infinitary Kripke-Platek set theory are realised. Finally, we use a variant of our notion of realisability to show that the propositional admissible rules of (finitary) intuitionistic Kripke-Platek set theory are exactly the admissible rules of intuitionistic propositional logic.     

Kripke‐Platek Set Theory and the Anti‐Foundation Axiom

The paper investigates the strength of the Anti‐Foundation Axiom, AFA, on the basis of Kripke‐Platek set theory without Foundation. It is shown that the addition of AFA considerably increases the proof theoretic strength.     

A note on Bar Induction in Constructive Set Theory

Bar Induction occupies a central place in Brouwerian mathematics. This note is concerned with the strength of Bar Induction on the basis of Constructive Zermelo‐Fraenkel Set Theory, CZF. It is shown that CZF augmented by decidable Bar Induction proves the 1‐consistency of CZF. This answers a question of P. Aczel who used Bar Induction to give a proof of the Lusin Separation Theorem in the constructive set theory CZF. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

BOOLEAN ALGEBRAS IN AST

In this paper we investigate Boolean algebras and their subalgebras in Alternative Set Theory (AST). We show that any two countable atomless Boolean algebras are isomorphic and we give an example of such a Boolean algebra. One other main result is, that there is an infinite Boolean algebra freely generated by a set. At the end of the paper we show that the sentence “There is no non-trivial free group which is a set” is consistent with AST.     

Localizing the axioms

We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to a transitive model of ZFC. LZFC consists of Loc(ZFC) plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All ${\Pi_2}We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to a transitive model of ZFC. LZFC consists of Loc(ZFC) plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All P₂{\Pi_2} consequences of ZFC are provable in LZFC. LZFC strongly extends Kripke-Platek (KP) set theory minus Δ₀-Collection and minus ?{\in} -induction scheme. ZFC+ “there is an inaccessible cardinal” proves the consistency of LZFC. In LZFC we focus on models rather than cardinals, a transitive model being considered as the analogue of an inaccessible cardinal. Pushing this analogy further we define α-Mahlo models and P₁¹{\Pi_1^1} -indescribable models, the latter being the analogues of weakly compact cardinals. Also localization axioms of the form Loc(ZFC+f){Loc({\rm ZFC}+\phi)} are considered and their global consequences are examined. Finally we introduce the concept of standard compact cardinal (in ZFC) and some standard compactness results are proved.     

The Steenrod Algebra and Theories Associated to Hopf Algebras

We show that the homotopy category of products of Z/p-Eilenberg–Mac Lane spaces is an -algebra which algebraically is determined by the Steenrod algebra considered as a Hopf algebra with unstable structure.     

A parametrised choice principle and Martin's conjecture on Blackwell determinacy

We define a parametrised choice principle PCP which (under the assumption of the Axiom of Blackwell Determinacy) is equivalent to the Axiom of Determinacy. PCP describes the difference between these two axioms and could serve as a means of proving Martin's conjecture on the equivalence of these axioms. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Brown–Peterson Cohomology from Morava K-Theory

We give some structure to the Brown–Peterson cohomology (or its p-completion) of a wide class of spaces. The class of spaces are those with Morava K-theory even-dimensional. We can say that the Brown–Peterson cohomology is even-dimensional (concentrated in even degrees) and is flat as a BP*-module for the category of finitely presented BP*(BP)-modules. At first glance this would seem to be a very restricted class of spaces but the world abounds with naturally occurring examples: Eilenberg-Mac Lane spaces, loops of finite Postnikov systems, classifying spaces of most finite groups whose Morava K-theory is known (including the symmetric groups), QS²ⁿ, BO(n), MO(n), BO, Im J, etc. We finish with an explicit algebraic construction of the Brown–Peterson cohomology of a product of Eilenberg–Mac Lane spaces and a general Künneth isomorphism applicable to our situation.     

CCC forcing and splitting reals

The results of this paper were motivated by a problem of Prikry who asked if it is relatively consistent with the usual axioms of set theory that every nontrivial ccc forcing adds a Cohen or a random real. A natural dividing line is into weakly distributive posets and those which add an unbounded real. In this paper I show that it is relatively consistent that every nonatomic weakly distributive ccc complete Boolean algebra is a Maharam algebra, i.e. carries a continuous strictly positive submeasure. This is deduced from theP-ideal dichotomy, a statement which was first formulated by Abraham and Todorcevic [AT] and later extended by Todorcevic [T]. As an immediate consequence of this and the proof of the consistency of theP-ideal dichotomy we obtain a ZFC result which says that every absolutely ccc weakly distributive complete Boolean algebra is a Maharam algebra. Using a previous theorem of Shelah [Sh1] it also follows that a modified Prikry conjecture holds in the context of Souslin forcing notions, i.e. every nonatomic ccc Souslin forcing either adds a Cohen real or its regular open algebra is a Maharam algebra. Finally, I also show that every nonatomic Maharam algebra adds a splitting real, i.e. a set of integers which neither contains nor is disjoint from an infinite set of integers in the ground model. It follows from the result of [AT] that it is consistent relative to the consistency of ZFC alone that every nonatomic weakly distributive ccc forcing adds a splitting real.     

Local set theory

In 1945, Eilenberg and MacLane introduced the new mathematical notion of category. Unfortunately, from the very beginning, category theory did not fit into the framework of either Zermelo—Fraenkel set theory or even von Neumann—Bernays—Gödel set-class theory. For this reason, in 1959, MacLane posed the general problem of constructing a new, more flexible, axiomatic set theory which would be an adequate logical basis for the whole of naïve category theory. In this paper, we give axiomatic foundations for local set theory. This theory might be one of the possible solutions of the MacLane problem.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 194–212.Original Russian Text Copyright © 2005 by V. K. Zakharov.This revised version was published online in April 2005 with a corrected issue number.