Zermelo-Fraenkel Research Articles

A Rigid Hyperfinite Type II1 Factor

Abstract We show that it is relatively consistent with Zermelo-Fraenkel set theory with the axiom of choice that there exists a hyperfinite type $\textrm{II}_1$-factor of density character $\aleph _1$ that is not isomorphic to its opposite, does not have any outer automorphisms, and has trivial fundamental group.

Cardinality of the sets of all bijections, injections and surjections

The results of Zarzycki for the cardinality of the sets of all bijections, surjections, and injections are generalized to the case when the domains and codomains are infinite and different. The elementary proofs the cardinality of the sets of bijections and surjections are given within the framework of the Zermelo-Fraenkel set theory with the axiom of choice. The case of the set of all injections is considered in detail and more explicit an expression is obtained when the Generalized Continuum Hypothesis is assumed.

Power Kripke–Platek set theory and the axiom of choice

Abstract While power Kripke–Platek set theory, ${\textbf{KP}}({\mathcal{P}})$, shares many properties with ordinary Kripke–Platek set theory, ${\textbf{KP}}$, in several ways it behaves quite differently from ${\textbf{KP}}$. This is perhaps most strikingly demonstrated by a result, due to Mathias, to the effect that adding the axiom of constructibility to ${\textbf{KP}}({\mathcal{P}})$ gives rise to a much stronger theory, whereas in the case of ${\textbf{KP}}$, the constructible hierarchy provides an inner model, so that ${\textbf{KP}}$ and ${\textbf{KP}}+V=L$ have the same strength. This paper will be concerned with the relationship between ${\textbf{KP}}({\mathcal{P}})$ and ${\textbf{KP}}({\mathcal{P}})$ plus the axiom of choice or even the global axiom of choice, $\textbf{AC}_{\tiny {global}}$. Since $L$ is the standard vehicle to furnish a model in which this axiom holds, the usual argument for demonstrating that the addition of ${\textbf{AC}}$ or $\textbf{AC}_{\tiny {global}}$ to ${\textbf{KP}}({\mathcal{P}})$ does not increase proof-theoretic strength does not apply in any obvious way. Among other tools, the paper uses techniques from ordinal analysis to show that ${\textbf{KP}}({\mathcal{P}})+\textbf{AC}_{\tiny {global}}$ has the same strength as ${\textbf{KP}}({\mathcal{P}})$, thereby answering a question of Mathias. Moreover, it is shown that ${\textbf{KP}}({\mathcal{P}})+\textbf{AC}_{\tiny {global}}$ is conservative over ${\textbf{KP}}({\mathcal{P}})$ for $\varPi ^1_4$ statements of analysis. The method of ordinal analysis for theories with power set was developed in an earlier paper. The technique allows one to compute witnessing information from infinitary proofs, providing bounds for the transfinite iterations of the power set operation that are provable in a theory. As the theory ${\textbf{KP}}({\mathcal{P}})+\textbf{AC}_{\tiny {global}}$ provides a very useful tool for defining models and realizability models of other theories that are hard to construct without access to a uniform selection mechanism, it is desirable to determine its exact proof-theoretic strength. This knowledge can for instance be used to determine the strength of Feferman’s operational set theory with power set operation as well as constructive Zermelo–Fraenkel set theory with the axiom of choice.

Equivariant ZFA and the foundations of nominal techniques

Abstract We give an accessible presentation to the foundations of nominal techniques, lying between Zermelo–Fraenkel set theory and Fraenkel–Mostowski set theory, which has several nice properties including being consistent with the Axiom of Choice. We give two presentations of equivariance, accompanied by detailed yet user-friendly discussions of its theory and application.

Quantum Mechanics, Formalization and the Cosmological Constant Problem

Based on formal arguments from Zermelo–Fraenkel set theory we develop the environment for explaining and resolving certain fundamental problems in physics. By these formal tools we show that any quantum system defined by an infinite dimensional Hilbert space of states interferes with the spacetime structure M. M and the quantum system both gain additional degrees of freedom, given by models of Zermelo–Fraenkel set theory. In particular, M develops the ground state where classical gravity vanishes. Quantum mechanics distinguishes set-theoretic random forcing such that M and gravitational degrees of freedom are parameterized by extended real line. The large scale smooth geometry compatible with the forcing extensions is one of exotic smoothness structures of $${\mathbb {R}}^4$$ . The amoeba forcing makes the old real line to have Lebesgue measure zero in the extended one. We apply the entire procedure to the cosmological constant problem, especially to discard the zero-modes contributions to the gravitational vacuum density. Moreover, there exists certain exotic smooth $${\mathbb {R}}^4$$ from which one determines the realistic, agreeing with observation, small value of the vacuum energy density.

Quantum Electrodynamics (QED) Renormalization is a Logical Paradox, Zeta Function Regularization is Logically Invalid, and Both are Mathematically Invalid

Quantum Electrodynamics (QED) renormalizaion is a paradox. It uses the Euler-Mascheroni constant, which is defined by a conditionally convergent series. But Riemann's series theorem proves that any conditionally convergent series can be rearranged to be divergent. This contradiction (a series that is both convergent and divergent) is a paradox in classical logic, intuitionistic logic, and Zermelo-Fraenkel set theory, and also contradicts the commutative and associative properties of addition. Therefore QED is mathematically invalid. Zeta function regularization equates two definitions of the Zeta function at domain values where they contradict (where the Dirichlet series definition is divergent and Riemann's definition is convergent). Doing so either creates a paradox (if Riemann's definition is true), or is logically invalid (if Riemann's definition is false). We show that Riemann's definition is false, because the derivation of Riemann's definition includes a contradiction: the use of both the Hankel contour and Cauchy's integral theorem. Also, a third definition of the Zeta function is proven to be false. The Zeta function has no zeros, so the Riemann hypothesis is a paradox, due to material implication and vacuous subjects.

Naturality and definability II

We regard an algebraic construction as a set-theoretically defined map taking structures A to structures B which have A as a distinguished part, in such a way that any isomorphism from A to A' lifts to an isomorphism from B to B'. In general the construction defines B up to isomorphism over A. A construction is uniformisable if the set-theoretic definition can be given in a form such that for each A the corresponding B is determined uniquely. A construction is natural if restriction from B to its part A always determines a map from the automorphism group of B to that of A which is a split surjective group homomorphism. We prove that there is no transitive model of ZFC (Zermelo-Fraenkel set theory with Choice) in which the uniformisable constructions are exactly the natural ones. We construct a transitive model of ZFC in which every uniformisable construction (with a restriction on the parameters in the formulas defining the construction) is ‘weakly’ natural. Corollaries are that the construction of algebraic closures of fields and the construction of divisible hulls of abelian groups have no uniformisations definable in ZFC without parameters.

Local External/Internal Symmetry of Smooth Manifolds and Lack of Tovariance in Physics

Category theory allows one to treat logic and set theory as internal to certain categories. What is internal to SET is 2-valued logic with classical Zermelo–Fraenkel set theory, while for general toposes it is typically intuitionistic logic and set theory. We extend symmetries of smooth manifolds with atlases defined in Set towards atlases with some of their local maps in a topos T . In the case of the Basel topos and R 4 , the local invariance with respect to the corresponding atlases implies exotic smoothness on R 4 . The smoothness structures do not refer directly to Casson handless or handle decompositions, which may be potentially useful for describing the so far merely putative exotic R 4 underlying an exotic S 4 (should it exist). The tovariance principle claims that (physical) theories should be invariant with respect to the choice of topos with natural numbers object and geometric morphisms changing the toposes. We show that the local T -invariance breaks tovariance even in the weaker sense.

NATURAL FORMALIZATION: DERIVING THE CANTOR-BERNSTEIN THEOREM IN ZF

AbstractNatural Formalization proposes a concrete way of expanding proof theory from the meta-mathematical investigation of formal theories to an examination of “the concept of the specifically mathematical proof.” Formal proofs play a role for this examination in as much as they reflect the essential structure and systematic construction of mathematical proofs. We emphasize three crucial features of our formal inference mechanism: (1) the underlying logical calculus is built for reasoning with gaps and for providing strategic directions, (2) the mathematical frame is a definitional extension of Zermelo–Fraenkel set theory and has a hierarchically organized structure of concepts and operations, and (3) the construction of formal proofs is deeply connected to the frame through rules for definitions and lemmas.To bring these general ideas to life, we examine, as a case study, proofs of the Cantor–Bernstein Theorem that do not appeal to the principle of choice. A thorough analysis of the multitude of “different” informal proofs seems to reduce them to exactly one. The natural formalization confirms that there is one proof, but that it comes in two variants due to Dedekind and Zermelo, respectively. In this way it enhances the conceptual understanding of the represented informal proofs. The formal, computational work is carried out with the proof search system AProS that serves as a proof assistant and implements the above inference mechanism; it can be fully inspected at http://www.phil.cmu.edu/legacy/Proof_Site/.We must—that is my conviction—take the concept of the specifically mathematical proof as an object of investigation.Hilbert 1918

Satisfiability is False Intuitionistically: A Question from Dana Scott

Satisfiability or Sat $$^{1}$$ is the metatheoretic statement Every formally intuitionistically consistent set of first-order sentences has a model. The models in question are the Tarskian relational structures familiar from standard first-order model theory (Chang and Keisler in Model theory, 3rd edn, Dover Publications Inc., Mineola, 2012), but here treated within intuitionistic metamathematics. We prove that both IZF, intuitionistic Zermelo–Fraenkel set theory, and HAS, second-order Heyting arithmetic, prove Sat $$^{1}$$ to be false outright. Following the lead of Carter (Notre Dame J Form Log 49(1):75–95, 2008), we then generalize this result to some provably intermediate first-order logics, including the Rose logic (Trans Am Math Soc 61:1–19, 1953). These metatheorems distinguish the intuitionistic foundational significance of Sat $$^{1}$$ sharply from that of Sat $$^{0}$$ , the satisfiability claim for intuitionistic propositional logic. At the same time, they establish intuitionistic connections with and between Test, COMP $$^{0}$$ , and $$\mathbf{COMP }^{1}$$ . Here, Test is the scheme of Testability, and COMP $$^{0}$$ and $$\mathbf{COMP }^{1}$$ are completeness for intuitionistic propositional logic and predicate logic, respectively.

Determinate logic and the Axiom of Choice

Takeuti introduced an infinitary proof system for determinate logic and showed that for transitive models of Zermelo-Fraenkel set theory with the Axiom of Dependent Choice that contain all reals, the cut-elimination theorem is equivalent to the Axiom of Determinacy, and in particular contradicts the Axiom of Choice.We consider variants of Takeuti's theorem without assuming the failure of the Axiom of Choice. For instance, we show that if one removes atomic formulae of infinite arity from the language of Takeuti's proof system, then cut elimination is equivalent to a determinacy hypothesis provable, e.g., in ZFC + “there are infinitely many Woodin cardinals.” A slight extension of the proof system admits cut elimination for countable sequents under the same assumptions.A simple modification of the proof system yields analogs of the results above for the Axiom of Real Determinacy and uncountably many Woodin cardinals.

Finitely Supported Sets Containing Infinite Uniformly Supported Subsets

The theory of finitely supported algebraic structures represents a reformulation of Zermelo-Fraenkel set theory in which every construction is finitely supported according to the action of a group of permutations of some basic elements named atoms. In this paper we study the properties of finitely supported sets that contain infinite uniformly supported subsets, as well as the properties of finitely supported sets that do not contain infinite uniformly supported subsets. For classical atomic sets, we study whether they contain or not infinite uniformly supported subsets.

The Ramsey property implies no mad families

We show that if all collections of infinite subsets of N have the Ramsey property, then there are no infinite maximal almost disjoint (mad) families. The implication is proved in Zermelo-Fraenkel set theory with only weak choice principles. This gives a positive solution to a long-standing problem that goes back to Mathias [A. R. D. Mathias, Ann. Math. Logic 12, 59-111 (1977)]. The proof exploits an idea which has its natural roots in ergodic theory, topological dynamics, and invariant descriptive set theory: We use that a certain function associated to a purported mad family is invariant under the equivalence relation [Formula: see text] and thus is constant on a "large" set. Furthermore, we announce a number of additional results about mad families relative to more complicated Borel ideals.

The entanglement of logic and set theory, constructively

ABSTRACTTheories of sets such as Zermelo Fraenkel set theory are usually presented as the combination of two distinct kinds of principles: logical and set-theoretic principles. The set-theoretic principles are imposed ‘on top’ of first-order logic. This is in agreement with a traditional view of logic as universally applicable and topic neutral. Such a view of logic has been rejected by the intuitionists, on the ground that quantification over infinite domains requires the use of intuitionistic rather than classical logic. In the following, I consider constructive set theories, which use intuitionistic rather than classical logic, and argue that they manifest a distinctive interdependence or an entanglement between sets and logic. In fact, Martin-Löf type theory identifies fundamental logical and set-theoretic notions. Remarkably, one of the motivations for this identification is the thought that classical quantification over infinite domains is problematic, while intuitionistic quantification is not. The approach to quantification adopted in Martin-Löf’s type theory is subtly interconnected with its predicativity. I conclude by recalling key aspects of an approach to predicativity inspired by Poincaré, which focuses on the issue of correct quantification over infinite domains and relate it back to Martin-Löf type theory.

Properties of the atoms in finitely supported structures

The goal of this paper is to present a collection of properties of the set of atoms and the set of finite injective tuples of atoms, as well as of the (finite and cofinite) powersets of atoms in the framework of finitely supported structures. Some properties of atoms are obtained by translating classical Zermelo–Fraenkel results into the new framework, but several important properties are specific to finitely supported structures (i.e. they do not have related classical Zermelo–Fraenkel and related non-atomic correspondents).

Foundations of Transmathematics

Transmathematics has the ambition to be a total mathematics. Many areas of the usual mathematics have been totalised in the transmathematics programme but the totalisations have all been carried out with the usual set theory, ZFC, Zermelo-Fraenkel set theory with the Axiom of Choice. This set theory is adequate but it is, itself, partial. Here we introduce a total set theory as a foundation for transmathematics.
 Surprisingly we adopt naive set theory. It is usually considered that the Russell Paradox demonstrates that naive set theory is incoherent because an apparently well-specified set, the Russell Set, cannot exist. We dissolve this paradox by showing that the specification of the Russell Set admits many unproblematical sets that do not contain themselves and, furthermore, unequivocally requires that the Russell Set does not contain itself because, were it to do so, that one element of the Russell Set would have contradictory membership. Having resolved the Russell Paradox, we go on to make the case that naive set theory is a paraconsistent logic.
 In order to demonstrate the sufficiency of naive set theory, as a basis for transmathematics, we introduce the transordinals. The von Neumann ordinals supply the usual ordinals, the simplest unordered set is identical to transreal nullity, and the Russell Set, excluding nullity, is the greatest ordinal, identical to transreal infinity. The generalisation of the transordinals to the whole of established transmathematics is already known.
 As naive set theory contains all other set theories, it provides a backwardly compatible foundation for the whole of mathematics.

The Urysohn Lemma is independent of ZF + Countable Choice

We prove that it is relatively consistent with Z F \mathsf {ZF} (i.e., Zermelo–Fraenkel set theory without the Axiom of Choice ( A C \mathsf {AC} )) that the Axiom of Countable Choice ( A C ℵ 0 \mathsf {AC}^{\aleph _{0}} ) is true, but the Urysohn Lemma ( U L \mathsf {UL} ), and hence the Tietze Extension Theorem ( T E T \mathsf {TET} ), is false. This settles the corresponding open problem in P. Howard and J. E. Rubin [Consequences of the Axiom of Choice, Mathematical Surveys and Monographs, Vol. 59, American Mathematical Society, Providence, RI, 1998]. We also prove that in Läuchli’s permutation model of Z F A \mathsf {ZFA} + + ¬ U L \neg \mathsf {UL} , A C ℵ 0 \mathsf {AC}^{\aleph _{0}} is false. This fills the gap in information in the above monograph of Howard and Rubin.

Embedding C*-Algebras Into the Calkin Algebra

Abstract We prove that, under Martin’s axiom, every ${\mathrm{C}}^\ast $-algebra of density character less than continuum embeds into the Calkin algebra. Furthermore, we show that it is consistent with Zermelo-Fraenkel set theory plus the axiom of choice, ZFC, that there is a $\mathrm{C}^\ast $-algebra of density character less than continuum that does not embed into the Calkin algebra.

AN EXTENSION OF A THEOREM OF ZERMELO

AbstractWe show that if $(M,{ \in _1},{ \in _2})$ satisfies the first-order Zermelo–Fraenkel axioms of set theory when the membership relation is ${ \in _1}$ and also when the membership relation is ${ \in _2}$, and in both cases the formulas are allowed to contain both ${ \in _1}$ and ${ \in _2}$, then $\left( {M, \in _1 } \right) \cong \left( {M, \in _2 } \right)$, and the isomorphism is definable in $(M,{ \in _1},{ \in _2})$. This extends Zermelo’s 1930 theorem in [6].

Preservation of choice principles under realizability

AbstractEspecially nice models of intuitionistic set theories are realizability models $V({\mathcal A})$, where $\mathcal A$ is an applicative structure or partial combinatory algebra. This paper is concerned with the preservation of various choice principles in $V({\mathcal A})$ if assumed in the underlying universe $V$, adopting Constructive Zermelo–Fraenkel as background theory for all of these investigations. Examples of choice principles are the axiom schemes of countable choice, dependent choice, relativized dependent choice and the presentation axiom. It is shown that any of these axioms holds in $V(\mathcal{A})$ for every applicative structure $\mathcal A$ if it holds in the background universe.1 It is also shown that a weak form of the countable axiom of choice, $\textbf{AC}^{\boldsymbol{\omega , \omega }}$, is rendered true in any $V(\mathcal{A})$ regardless of whether it holds in the background universe. The paper extends work by McCarty (1984, Realizability and Recursive Mathematics, PhD Thesis) and Rathjen (2006, Realizability for constructive Zermelo–Fraenkel set theory. In Logic Colloquium 03, pp. 282–314).