Zermelo-Fraenkel Research Articles

A formalised theorem in the partition calculus

A paper on ordinal partitions by Erdős and Milner [7] has been formalised using the proof assistant Isabelle/HOL, augmented with a library for Zermelo–Fraenkel set theory. The work is part of a project on formalising the partition calculus. The chosen material is particularly appropriate in view of the substantial corrections [8] later published by its authors, illustrating the potential value of formal verification.

Should Type Theory Replace Set Theory as the Foundation of Mathematics?

Mathematicians often consider Zermelo-Fraenkel Set Theory with Choice (ZFC) as the only foundation of Mathematics, and frequently don’t actually want to think much about foundations. We argue here that modern Type Theory, i.e. Homotopy Type Theory (HoTT), is a preferable and should be considered as an alternative.

Coq Formalization of ZFC Set Theory for Teaching Scenarios

Coq Formalization of ZFC Set Theory for Teaching Scenarios

Sequential and distributive forcings without choice

AbstractIn the Zermelo–Fraenkel set theory with the Axiom of Choice, a forcing notion is “ $\kappa $ -distributive” if and only if it is “ $\kappa $ -sequential.” We show that without the Axiom of Choice, this equivalence fails, even if we include a weak form of the Axiom of Choice, the Principle of Dependent Choice for $\kappa $ . Still, the equivalence may still hold along with very strong failures of the Axiom of Choice, assuming the consistency of large cardinal axioms. We also prove that although a $\kappa $ -distributive forcing notion may violate Dependent Choice, it must preserve the Axiom of Choice for families of size $\kappa $ . On the other hand, a $\kappa $ -sequential can violate the Axiom of Choice for countable families. We also provide a condition of “quasiproperness” which is sufficient for the preservation of Dependent Choice, and is also necessary if the forcing notion is sequential.

Sets, Properties and Truth Values: A Category-Theoretic Approach to Zermelo’s Axiom of Separation

In 1908 the German mathematician Ernst Zermelo gave an axiomatization of the concept of set. His axioms remain at the core of what became to be known as Zermelo-Fraenkel set theory. There were two axioms that received diverse criticisms at the time: the axiom of choice and the axiom of separation. This paper centers around one question this latter axiom raised. The main purpose is to show how this question might be solved with the aid of another, more recent mathematical theory of sets which, like Zermelo’s, has numerous philosophical underpinnings. Keywords: properties of sets, foundations of mathematics, axiom of separation, subobject classifier, truth values

The Paradox of Classical Reasoning

Intuitively, the more powerful a theory is, the greater the variety and quantity of ideas can be expressed through its formal language. Therefore, when comparing two theories concerning the same subject, it seems only reasonable to compare the expressive powers of their formal languages. On condition that the quantum mechanical description is universal and so can be applied to macroscopic systems, quantum theory is required to be more powerful than classical mechanics. This implies that the formal language of Hilbert space theory must be more expressive than that of Zermelo-Fraenkel set theory (the language of classical formalism). However, as shown in the paper, such a requirement cannot be met. As a result, classical and quantum formalisms cannot be in a hierarchical relation, that is, include one another. This fact puts in doubt the quantum-classical correspondence and undermines the reductionist approach to the physical world.

A Model in Which Well-Orderings of the Reals Appear at a Given Projective Level

The problem of the existence of analytically definable well-orderings at a given level of the projective hierarchy is considered. This problem is important as a part of the general problem of the study of the projective hierarchy in the ongoing development of descriptive set theory. We make use of a finite support product of the Jensen-type forcing notions to define a model of set theory ZFC in which, for a given n>2, there exists a good Δn1 well-ordering of the reals but there are no such well-orderings in the class Δn−11. Therefore the existence of a well-ordering of the reals at a certain level n>2 of the projective hierarchy does not imply the existence of such a well-ordering at the previous level n−1. This is a new result in such a generality (with n>2 arbitrary), and it may lead to further progress in studies of the projective hierarchy.

EXTENSIONAL REALIZABILITY AND CHOICE FOR DEPENDENT TYPES IN INTUITIONISTIC SET THEORY

AbstractIn [17], we introduced an extensional variant of generic realizability [22], where realizers act extensionally on realizers, and showed that this form of realizability provides inner models of $\mathsf {CZF}$ (constructive Zermelo–Fraenkel set theory) and $\mathsf {IZF}$ (intuitionistic Zermelo–Fraenkel set theory), that further validate $\mathsf {AC}_{\mathsf {FT}}$ (the axiom of choice in all finite types). In this paper, we show that extensional generic realizability validates several choice principles for dependent types, all exceeding $\mathsf {AC}_{\mathsf {FT}}$ . We then show that adding such choice principles does not change the arithmetic part of either $\mathsf {CZF}$ or $\mathsf {IZF}$ .

Random World and Quantum Mechanics

Quantum mechanics (QM) predicts probabilities on the fundamental level which are, via Born probability law, connected to the formal randomness of infinite sequences of QM outcomes. Recently it has been shown that QM is algorithmic 1-random in the sense of Martin–Löf. We extend this result and demonstrate that QM is algorithmic $$\omega$$ -random and generic, precisely as described by the ’miniaturisation’ of the Solovay forcing to arithmetic. This is extended further to the result that QM becomes Zermelo–Fraenkel Solovay random on infinite-dimensional Hilbert spaces. Moreover, it is more likely that there exists a standard transitive ZFC model M, where QM is expressed in reality, than in the universe V of sets. Then every generic quantum measurement adds to M the infinite sequence, i.e. random real $$r\in 2^{\omega }$$ , and the model undergoes random forcing extensions M[r]. The entire process of forcing becomes the structural ingredient of QM and parallels similar constructions applied to spacetime in the quantum limit, therefore showing the structural resemblance of both in this limit. We discuss several questions regarding measurability and possible practical applications of the extended Solovay randomness of QM. The method applied is the formalization based on models of ZFC; however, this is particularly well-suited technique to recognising randomness questions of QM. When one works in a constant model of ZFC or in axiomatic ZFC itself, the issues considered here remain hidden to a great extent.

Set theoretical analogues of the Barwise-Schlipf theorem

We prove the following characterizations of nonstandard models of ZFC (Zermelo-Fraenkel set theory with the axiom of choice) that have an expansion to a model of GB (Gödel-Bernays class theory) plus Δ11-CA (the scheme of Δ11-Comprehension). In what follows, M(α):=(V(α),∈)M, LM is the set of formulae of the infinitary logic L∞,ω that appear in the well-founded part of M, and Σ11-AC is the scheme of Σ11-Choice.Theorem A.The following are equivalent for a nonstandard modelMof ZFC of any cardinality:(a)M(α)≺LMMfor an unbounded collection ofα∈OrdM.(b)(M,X)⊨GB+Δ11-CA, whereXis the family ofLM-definable subsets ofM.(c)There isXsuch that(M,X)⊨GB+Δ11-CA.Theorem B.The following are equivalent for a countable nonstandard model of ZFC:(a)M(α)≺LMMfor an unbounded collection ofα∈OrdM.(b)There isXsuch that(M,X)⊨GB+Δ11-CA+Σ11-AC.

The cardinality of the partitions of a set in the absence of the Axiom of Choice

Abstract In the Zermelo–Fraenkel set theory (ZF), $|\textrm {fin}(A)|<2^{|A|}\leq |\textrm {Part}(A)|$ for any infinite set $A$, where $\textrm {fin}(A)$ is the set of finite subsets of $A$, $2^{|A|}$ is the cardinality of the power set of $A$ and $\textrm {Part}(A)$ is the set of partitions of $A$. In this paper, we show in ZF that $|\textrm {fin}(A)|<|\textrm {Part}_{\textrm {fin}}(A)|$ for any set $A$ with $|A|\geq 5$, where $\textrm {Part}_{\textrm {fin}}(A)$ is the set of partitions of $A$ whose members are finite. We also show that, without the Axiom of Choice, any relationship between $|\textrm {Part}_{\textrm {fin}}(A)|$ and $2^{|A|}$ for an arbitrary infinite set $A$ cannot be concluded.

Provability logic: models within models in Peano Arithmetic

In 1994 Jech gave a model-theoretic proof of Gödel’s second incompleteness theorem for Zermelo–Fraenkel set theory in the following form: {{,mathrm{mathrm {ZF}},}} does not prove that {{,mathrm{mathrm {ZF}},}} has a model. Kotlarski showed that Jech’s proof can be adapted to Peano Arithmetic with the role of models being taken by complete consistent extensions. In this note we take another step in the direction of replacing proof-theoretic by model-theoretic arguments. We show, without the need of formalizing the proof of the completeness theorem within {{,mathrm{mathrm {PA}},}}, that the existence of a model of {{,mathrm{mathrm {PA}},}} of complexity Sigma ^0_2 is independent of {{,mathrm{mathrm {PA}},}}, where a model is identified with the set of formulas with parameters which hold in the model. Our approach is based on a new interpretation of the provability logic of Peano Arithmetic where Box phi is defined as the formalization of “phi is true in every Sigma ^0_2-model”.

Representation and Spacetime: The Hole Argument Revisited

ABSTRACT Ladyman and Presnell have recently argued that the Hole argument is naturally resolved when spacetime is represented within homotopy type theory rather than set theory. The core idea behind their proposal is that the argument does not confront us with any indeterminism, since the set-theoretically different representations of spacetime involved in the argument are homotopy type-theoretically identical. In this article, we will offer a new resolution based on ZFC set theory to the argument. It neither relies on a constructive-intuitionistic form of mathematics, as used by Ladyman and Presnell, nor is foundationally problematic, such as the existing set-theoretic suggestions.

Representations and the Foundations of Mathematics

The representation of mathematical objects in terms of (more) basic ones is part and parcel of (the foundations of) mathematics. In the usual foundations of mathematics, i.e. $\textsf{ZFC}$ set theory, all mathematical objects are represented by sets, while ordinary, i.e. non-set theoretic, mathematics is represented in the more parsimonious language of second-order arithmetic. This paper deals with the latter representation for the rather basic case of continuous functions on the reals and Baire space. We show that the logical strength of basic theorems named after Tietze, Heine, and Weierstrass, changes significantly upon the replacement of 'second-order representations' to 'third-order functions'. We discuss the implications and connections to the Reverse Mathematics program and its foundational claims regarding predicativist mathematics and Hilbert's program for the foundations of mathematics. Finally, we identify the problem caused by representations of continuous functions and formulate a criterion to avoid problematic codings within the bigger picture of representations.

THE DISCONTINUITY PROBLEM

AbstractMatthias Schröder has asked the question whether there is a weakest discontinuous problem in the topological version of the Weihrauch lattice. Such a problem can be considered as the weakest unsolvable problem. We introduce the discontinuity problem, and we show that it is reducible exactly to the effectively discontinuous problems, defined in a suitable way. However, in which sense this answers Schröder’s question sensitively depends on the axiomatic framework that is chosen, and it is a positive answer if we work in Zermelo–Fraenkel set theory with dependent choice and the axiom of determinacy $\mathsf {AD}$ . On the other hand, using the full axiom of choice, one can construct problems which are discontinuous, but not effectively so. Hence, the exact situation at the bottom of the Weihrauch lattice sensitively depends on the axiomatic setting that we choose. We prove our result using a variant of Wadge games for mathematical problems. While the existence of a winning strategy for Player II characterizes continuity of the problem (as already shown by Nobrega and Pauly), the existence of a winning strategy for Player I characterizes effective discontinuity of the problem. By Weihrauch determinacy we understand the condition that every problem is either continuous or effectively discontinuous. This notion of determinacy is a fairly strong notion, as it is not only implied by the axiom of determinacy $\mathsf {AD}$ , but it also implies Wadge determinacy. We close with a brief discussion of generalized notions of productivity.

Preference Change and Utility Conditionalization

Olav Vassend has recently (2021) presented a decision-theoretic argument for updating utility functions by what he calls “utility conditionalization.” Vassend’s argument is meant to mirror closely the well-known argument for Bayesian conditionalization due to Hilary Greaves and David Wallace (2006). I show that Vassend’s argument is inconsistent with ZF set theory and argue that it therefore does not provide support for utility conditionalization.

The Logical Study of Non-Well-Founded Set and Circulation Phenomenon

In recent years, since a great deal of circular phenomenon, there has been a furry of interest in them. To explain various circular phenomenon, the study of set theory extended well -founded sets to non-well-founded set. Based on this basis, the paper discusses the logical theoretical basis of circular phenomena. Non-well-founded set theory ZFA allows primitive existence. Primitive is an object that has no elements and is not a class in itself. It is based on the set theory ZFC after the axiom of foundation FA is removed, and the anti-basic axiom AFA is added to ZFC. ZFC here refers to ZF set theory with axiom of choice. According to axiomatic set theory, for ZFC's regular axioms, the set in its universe is a well set. If the regular axiom is removed, and the infinite decline is allowed to belong to the relational chain, then the non-well-founded set can be introduced. Firstly, this paper introduces the basic concept of non-well-founded set, the foundation axiom and the anti-founded axioms. Secondly, we dicusses the limit of the foundation axiom. Thirdly, we exhibit the history and present situation of the research on non-well-founded sets are briefly reviewed. Finally, the applications of non-well-founded sets in philosophy, linguistics, computer science, economics and many other fields is discussed. Because non-well-founded set theory will provide a better tool for dealing with circular phenomena naturally, it can be argued that circle is not vicious.

On the equivalence of Rudin's Lemma and the Boolean prime ideal theorem

Rudin's Lemma, which appeared more than forty years ago, is a consequence of the Axiom of Choice and has been playing fundamental roles in the study of quasicontinuity in domain theory. This note aims to revealing the fact that Rudin's Lemma (and its variants) and the Boolean prime ideal theorem are equivalent in Zermelo-Fraenkel Set Theory, which is choiceless Set Theory. Our main techniques rely on Erné's Separation Lemma for Locales and a variant of Rado's Selection Principle due to Cowen.

On a geometric statement of Ramsey type

Abstract A simple geometric assertion of Ramsey type, concerning families of straight lines in the Euclidean space R 3 \mathbb{R}^{3} , is formulated, and it is shown that the assertion turns out to be undecidable within the framework of ZFC set theory.

Formalizing Ordinal Partition Relations Using Isabelle/HOL

ABSTRACT This is an overview of a formalization project in the proof assistant Isabelle/HOL of a number of research results in infinitary combinatorics and set theory (more specifically in ordinal partition relations) by Erdős–Milner, Specker, Larson and Nash-Williams, leading to Larson’s proof of the unpublished result by E.C. Milner asserting that for all , . This material has been recently formalised by Paulson and is available on the Archive of Formal Proofs; here we discuss some of the most challenging aspects of the formalization process. This project is also a demonstration of working with Zermelo–Fraenkel set theory in higher-order logic.