Zermelo-Fraenkel Research Articles

Representable topologies and locally connected spaces

The objective of this paper is to study certain order-theoretic properties of locally connected topologies. In the main theorem we prove that the assertion that a locally connected topology which satisfies the countable chain condition has the continuous representability property is undecidable in ZFC set theory. Some related problems and generalizations are also included.

Binary Refinement Implies Discrete Exponentiation

Working in the weakening of constructive Zermelo-Fraenkel set theory in which the subset collection scheme is omitted, we show that the binary re.nement principle implies all the instances of the exponentiation axiom in which the basis is a discrete set. In particular binary re.nement implies that the class of detachable subsets of a set form a set. Binary re.nement was originally extracted from the fullness axiom, an equivalent of subset collection, as a principle that was su.cient to prove that the Dedekind reals form a set. Here we show that the Cauchy reals also form a set. More generally, binary refinement ensures that one remains in the realm of sets when one starts from discrete sets and one applies the operations of exponentiation and binary product a finite number of times.

Choiceless, Pointless, but not Useless: Dualities for Preframes

We provide the appropriate common ‘(pre)framework’ for various central results of domain theory and topology, like the Lawson duality of continuous domains, the Hofmann–Lawson duality between continuous frames and locally compact sober spaces, the Hofmann–Mislove theorems about continuous semilattices of compact saturated sets, or the theory of stably continuous frames and their topological manifestations. Suitable objects for the pointfree approach are quasiframes, i.e., up-complete meet-semilattices with top, and preframes, i.e., meet-continuous quasiframes. We introduce the pointfree notion of locally compact well-filtered preframes, show that they are just the continuous preframes (using a slightly modified definition of continuity) and establish several natural dualities for the involved categories. Moreover, we obtain various characterizations of preframes having duality. Our results hold in ZF set theory without any choice principles.

Local theory of sets as a foundation for category theory and its connection with the Zermelo-Fraenkel set theory

Local theory of sets as a foundation for category theory and its connection with the Zermelo-Fraenkel set theory

Ordinal computations

The notion of ordinal computability is defined by generalising standard Turing computability on tapes of length . In this paper we present a new proof of this theorem that makes use of a theory SO axiomatising the class of sets of ordinals in a model of set theory. The theory SO and the standard Zermelo–Fraenkel axiom system ZFC can be canonically interpreted in each other. The proof of the fundamental theorem is based on showing that the class of sets that are ordinal computable from ordinal parameters forms a model of SO.

Comprehending software correctness implies comprehending an intelligence-related limitation

This article applies mathematical logic to obtain a rigorous foundation for previous inherently nonrigorous results and also extends those previous results. Roughly speaking, our main theorem states: any agent A that comprehends the correctness-related properties of software S also comprehends an intelligence-related limitation of S . The theorem treats the output of S , if any, as an attempt at solving a halting problem. Previous nonrigorous attempts to obtain similar theorems depend on infallibility assumptions on both the agent and the software. The hypothesis that intelligent agents and intelligent software must be infallible has been widely questioned. In addition, recent work by others has determined that well-known previous attempts use a fallacious form of reasoning; that is, the same form of reasoning can yield paradoxical results. Our main theorem avoids infallibility assumptions on both the agent and the software. In addition, our proof is rigorous, in the sense that in principle one can carry it out in Zermelo-Fraenkel set theory. The software correctness framework considered in the main theorem is that of Hoare logic.

Burgess on Plural Logic and Set Theory

John Burgess in a 2004 paper combined plural logic and a new version of the idea of limitation of size to give an elegant motivation of the axioms of ZFC set theory. His proposal is meant to improve on earlier work by Paul Bernays in two ways. I argue that both attempted improvements fail.

What is Neologicism?

Logicism is a thesis about the foundations of mathematics, roughly, that mathematics is derivable from logic alone. It is now widely accepted that the thesis is false and that the logicist program of the early 20th century was unsuccessful. Frege's [1893/1903] system was inconsistent and the Whitehead and Russell [1910–1913] system was not thought to be logic, given its axioms of infinity, reducibility, and choice. Moreover, both forms of logicism are in some sense non-starters, since each asserts the existence of objects (courses of values, propositional functions, etc.), something which many philosophers think logic is not supposed to do. Indeed, the tension in the idea underlying logicism, that the axioms and theorems of mathematics can be derived as theorems of logic, is obvious: on the one hand, there are numerous existence claims among the theorems of mathematics, while on the other, it is thought to be impossible to prove the existence of anything from logic alone. According to one well-received view, logicism was replaced by a very different account of the foundations of mathematics, in which mathematics was seen as the study of axioms and their consequences in models consisting of the sets described by Zermelo-Fraenkel set theory (ZF). Mathematics, on this view, is just applied set theory.

CARDINAL INVARIANTS AND EVENTUALLY DIFFERENT FUNCTIONS

In this paper we study several kinds of maximal almost disjoint families. In the main result of this paper we show that for successor cardinals κ, there is an unexpected connection between invariants ae(κ), b(κ) and a certain cardinal invariant md(κ+) on κ+. As a corollary we get for example the following result. For a successor cardinal κ, even assuming that κ<κ = κ and 2κ = κ+, the following is not provable in Zermelo–Fraenkel set theory. There is a κ+-cc poset which does not collapse κ and which forces a(κ) = κ+ < ae(κ) = κ++ = 2κ. We also apply the ideas from the proofs of these results to study a = a(ω) and non(M). 2000 Mathematics Subject Classification 03E17 (primary), 03E05 (secondary).

CARDINAL INVARIANTS AND EVENTUALLY DIFFERENT FUNCTIONS

In this paper we study several kinds of maximal almost disjoint families. In the main result of this paper we show that for successor cardinals κ, there is an unexpected connection between invariants ae(κ), b(κ) and a certain cardinal invariant md(κ+) on κ+. As a corollary we get for example the following result. For a successor cardinal κ, even assuming that κ<κ = κ and 2κ = κ+, the following is not provable in Zermelo–Fraenkel set theory. There is a κ+-cc poset which does not collapse κ and which forces a(κ) = κ+ < ae(κ) = κ++ = 2κ. We also apply the ideas from the proofs of these results to study a = a(ω) and non(M). 2000 Mathematics Subject Classification 03E17 (primary), 03E05 (secondary).

Characterizing the interpretation of set theory in Martin-Löf type theory

Constructive Zermelo–Fraenkel set theory, CZF, can be interpreted in Martin-Löf type theory via the so-called propositions-as-types interpretation. However, this interpretation validates more than what is provable in CZF. We now ask ourselves: is there a reasonably simple axiomatization (by a few axiom schemata say) of the set-theoretic formulae validated in Martin-Löf type theory? The answer is yes for a large collection of statements called the mathematical formulae. The validated mathematical formulae can be axiomatized by suitable forms of the axiom of choice. The paper builds on a self-interpretation of CZF (developed in [M. Rathjen, The formulae-as-classes interpretation of constructive set theory, in: Proof Technology and Computation (Proceedings of the International Summer School Marktoberdorf 2003) IOS Press, Amsterdam, 2004 (in press)]) that provides an “inner” model of CZF which also validates the so-called Π Σ -axiom of choice, ΠΣ – AC . The crucial technical step taken in the present paper is to investigate the absoluteness properties of this model under the hypothesis ΠΣ – AC . It is also shown that CZF plus the Π Σ -axiom of choice possesses the disjunction property, the numerical existence property and the existence property for an important group of formulae.

Model Approach to Nonstandard Analysis in the Context of Axiomatic Set Theory

In this paper, the model approach to nonstandard analysis is developed on the basis of Zermelo—Fraenkel axiomatic set theory with atoms. The traditional consideration of the standard superstructure V as the primary object of nonstandard analysis is justified. Set-theoretic axioms for the nonstandard system *V are obtained.

The disjunction and related properties for constructive Zermelo-Fraenkel set theory

AbstractThis paper proves that the disjunction property, the numerical existence property. Church's rule, and several other metamathematical properties hold true for Constructive Zermelo-Fraenkel Set Theory, CZF, and also for the theory CZF augmented by the Regular Extension Axiom.As regards the proof technique, it features a self-validating semantics for CZF that combines realizability for extensional set theory and truth.

An automated prover for Zermelo–Fraenkel set theory in Theorema

This paper presents some fundamental aspects of the design and the implementation of an automated prover for Zermelo–Fraenkel set theory within the Theorema system. The method applies the “Prove–Compute–Solve” paradigm as its major strategy for generating proofs in a natural style for statements involving constructs from set theory.

Properties of the real line and weak forms of the Axiom of Choice

We investigate, within the framework of Zermelo-Fraenkel set theory ZF, the interrelations between weak forms of the Axiom of Choice AC restricted to sets of reals. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

On constructing completions

AbstractThe Dedekind cuts in an ordered set form a set in the sense of constructive Zermelo–Fraenkel set theory. We deduce this statement from the principle of refinement, which we distill before from the axiom of fullness. Together with exponentiation, refinement is equivalent to fullness. None of the defining properties of an ordering is needed, and only refinement for two–element coverings is used.In particular, the Dedekind reals form a set: whence we have also refined an earlier result by Aczel and Rathjen, who invoked the full form of fullness. To further generalise this, we look at Richman's method to complete an arbitrary metric space without sequences, which he designed to avoid countable choice. The completion of a separable metric space turns out to be a set even if the original space is a proper class: in particular, every complete separable metric space automatically is a set.

Graph-Coloring and Choice a Note on a Note by Shelah and Soifer

Sharpening results of Shelah and Soifer we will show that in ZF set theory the Shelah-Soifer Graph G has the following properties: 1. If AC(2), the axiom of choice for families of 2-element sets, or AC(R), the axiom of choice for families of non-empty subsets of R, hold, then G is 2-colorable. 2. For every cardinal c and for every c-coloring f : G → c each of the coloring sets f −1(i) is either non-measurable or of measure zero. 3. For every 2-coloring f : G → 2 both coloring sets f −1(0) and f −1(1) are non-measurable. 4. For every n-coloring f : G → n where n < ℵ0, there exists a non-measurable coloring set f −1(i). 5. If CC(R), the axiom of choice for countable families of non-empty subsets of R, holds, then for every ℵ0-coloring f : G → ℵ0 there exists a non-measurable coloring set f −1(i). 6. If AD, the axiom of determinateness, holds then, for any cardinal c ≤ ℵ0, the graph G has no c-coloring.

Heyting-valued interpretations for Constructive Set Theory

We define and investigate Heyting-valued interpretations for Constructive Zermelo–Frankel set theory (CZF). These interpretations provide models for CZF that are analogous to Boolean-valued models for ZF and to Heyting-valued models for IZF. Heyting-valued interpretations are defined here using set-generated frames and formal topologies. As applications of Heyting-valued interpretations, we present a relative consistency result and an independence proof.

Largest fixed points of set continuous operators and Boffa's Anti-Foundation

In Aczel [1], the existence of largest (written “greatest” in Barwise and Moss [2]) fixed points of set continuous operators is proved assuming the schema version of dependent choices in Zermelo-Fraenkel set theory without the axiom of Foundation. In the present paper, we study whether the existence of largest fixed points of set continuous operators is provable without the schema version of dependent choices, using Boffa's weak antifoundation axioms. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Weak axioms of choice for metric spaces

In the framework of ZF, i.e., Zermelo-Fraenkel set theory without the axiom of choice AC, we show that if the family of all non-empty, closed subsets of a metric space ( X , d ) (X,d) has a choice function, then so does the family of all non-empty, open subsets of X X . In addition, we establish that the converse is not provable in ZF. We also show that the statement “every subspace of the real line R \mathbb {R} with the standard topology has a choice function for its family of all closed, non-empty subsets" is equivalent to the weak choice form “every continuum sized family of non-empty subsets of reals has a choice function".