Zermelo-Fraenkel Research Articles

Von Rimscha's Transitivity Conditions

In Zermelo-Fraenkel set theory with the axiom of choice every set has the same cardinal number as some ordinal. Von Rimscha has weakened this condition to “Every set has the same cardinal number as some transitive set”. In set theory without the axiom of choice, we study the deductive strength of this and similar statements introduced by von Rimscha.

Contrary Statements About Mathematics

The preceding paper ‘Strong statements of analysis’ by A. R. D. Mathias defends a so-called full-blooded set theory without full detail [3]. He again objects to a weak set theory which he calls ‘Mac’, in which the usual Zermelo–Fraenkel separation scheme is required only for formulas with suitably ‘restricted’ quantifiers. I had proposed that such separation is adequate for all standard uses of set theory in mathematics. But Mathias has not produced any counter examples of actual mathematics which requires the use of a stronger separation.

The Hahn representation theorem for ℓ-groups in ZFA

In [7] the author discussed the relative force —in the set theory ZF— of some representation theorems for ℓ-groups (lattice-ordered groups). One of the theorems not discussed in that paper is the Hahn representation theorem for abelian ℓ-groups. This result, originally proved by Hahn (see [8]) for totally ordered groups and half a century later by Conrad, Harvey and Holland for the general case (see [4]), states that any abelian ℓ-group can be embedded in a Hahn product of copies of R (the real line with its natural totally-ordered group structure). Both proofs rely heavily on Zorn's Lemma which is equivalent to AC (the axiom of choice).The aim of this work is to point out the use of non-constructible axioms (i.e., AC and weaker forms of it) in the proofs. Working in the frame of ZFA, that is, the Zermelo-Fraenkel set theory where a non-empty set of atoms is allowed, we present alternative proofs which, in the totally ordered case, do not require the use of AC. For basic concepts and notation on ℓ-groups the reader can refer to [1] and [2]. For set theory, to [11].

Zermelo and the Skolem Paradox

On October 4, 1937, Zermelo composed a small note entitled “Der Relativismus in der Mengenlehre und der sogenannte Skolemsche Satz”(“Relativism in Set Theory and the So-Called Theorem of Skolem”) in which he gives a refutation of “Skolem's paradox”, i.e., the fact that Zermelo-Fraenkel set theory—guaranteeing the existence of uncountably many sets—has a countable model. Compared with what he wished to disprove, the argument fails. However, at a second glance, it strongly documents his view of mathematics as based on a world of objects that could only be grasped adequately by infinitary means. So the refutation might serve as a final clue to his epistemological credo.Whereas the Skolem paradox was to raise a lot of concern in the twenties and the early thirties, it seemed to have been settled by the time Zermelo wrote his paper, namely in favour of Skolem's approach, thus also accepting the noncategoricity and incompleteness of the first-order axiom systems. So the paper might be considered a late-comer in a community of logicians and set theorists who mainly followed finitary conceptions, in particular emphasizing the role of first-order logic (cf. [8]). However, Zermelo never shared this viewpoint: In his first letter to Gödel of September 21, 1931, (cf. [1]) he had written that the Skolem paradox rested on the erroneous assumption that every mathematically definable notion should be expressible by a finite combination of signs, whereas a reasonable metamathematics would only be possible after this “finitistic prejudice” would have been overcome, “a task I have made my particular duty”.

A Single Axiom for Set Theory

Axioms in set theory typically have the form $\forall z \exists y\forall x(x \in y \leftrightarrow F x z )$, where $F$ is a relation which links $x$ with $z$ in some way. In this paper we introduce a particular linkage relation $L$ and a single axiom based on $L$ from which all the axioms of $\mathrm{Z}$ (Zermelo set theory) can be derived as theorems. The single axiom is presented both in informal and formal versions. This calls for some discussion of pertinent features of formal and informal axiomatic method and some discussion of pertinent features of the system $\mathrm{S}$ of set theory to be erected on the single axiom. $\mathrm{S}$ is shown to be somewhat stronger than $\mathrm{Z}$, but much weaker than $\mathrm{ZF}$ (Zermelo-Fraenkel set theory).

Strictly complete regularity of the Lawson topology on a continuous poset

For a continuous poset P, let ω(P) and λ(P) be the lower topology and the Lawson topology on P, respectively. In this paper, we constructively prove in the set theory ZFDC ω (the theory obtained by adjoining the axiom of ω dependent choices to the Zermelo–Fraenkel set theory) that if all lower closed subsets in (P,λ(P)) are closed in (P,ω(P)), then (P,λ(P)) is a strictly completely regular ordered space.

INDISTINGUISHABLE PARTICLES AND HIDDEN VARIABLES: PART II

In a previous work it was shown that it is possible to deal with collections of indistinguishable elementary particles in a set-theoretical framework, by using hidden variables. We propose in the present paper a set-theoretical axiomatics for collections of indiscernibles with no explicit mention to hidden variables. We also show, in this context, the fundamental role of the (micro) state in the process of individuation of classical and quantum particles. Finally, we discuss the importance of the axiom of choice in Zermelo-Fraenkel set theory in the context of quantum distributions of bosons and fermions.

Elementary Particles, Hidden Variables, And Hidden Predicates

We recently showed that it is possible to deal withcollections of indistinguishable elementary particles (in thecontext of quantum mechanics) in a set-theoretical framework, byusing hidden variables. We propose in the presentpaper another axiomatics for collections of indiscernibleswithout hidden variables, where hidden predicates are implicitlyassumed. We also discuss the possibility of a quasi-settheoretical picture for quantum theory. Quasi-set theory, basedon Zermelo-Fraenkel set theory, was developed for dealing withcollections of indistinguishable, but, not identical objects.

Revising Z: Part I – logic and semantics

Abstract. This is the first of two related papers. We introduce a simple specification logic Z C comprising a logic and a semantics (in ZF set theory) within which the logic is sound. We then provide an interpretation for (a rational reconstruction of) the specification language Z within Z C . As a result we obtain a sound logic for Z, including a basic schema calculus.

Revising Z: Part II – logical development

Abstract. This is the second of two related papers. In “Revising Z: Part I - logic and semantics” (this journal) we introduced a simple specification logic Z C comprising a logic and a semantics (in ZF set theory). We then provided an interpretation for (a rational reconstruction of) the specification language Z within Z C . As a result we obtained a sound logic for Z, including the basic schema calculus. In this paper we extend the basic framework with more sophisticated features (including schema operations) and we mount a critique of a number of concepts used in Z. We further demonstrate that the complications and confusions which these concepts introduce can be avoided without compromising expressibility.

No elementary embedding from v into v is definable from parameters

In 1970, Kenneth Kunen showed that there is no non-trivial elementary embedding of the universe V into itself [2] using the axiom of choice. Kunen remarked in his paper that the result can be formalized in Morse-Kelley theory of sets and classes. In this paper, we will work within ZF, Zermelo-Fraenkel axioms, and deal with embeddings definable with a formula and a parameter.In ZF, a “class” is usually synonymous with “property”, that is a class definable with a parameter, C = {x: φ(x,p)}, where φ is a formula in the language [∈}. Using this convention, let j be a class. Then “j is an elementary embedding of V into V” is not a single statement but a schema of statements “j preserves ψ” for each formula ψ. We prove that this schema is expressible in the language {∈} by a single formula:Lemma. An embedding j: V → V is elementary iff j preservesψ.Here ψ(α, ψ, a) is the property “a is an ordinal, φ is a formula and Vα.”The lemma is of course a schema of lemmas, one for each formula denning j and for each ψ to be preserved.Using this we prove our theorem in ZF (again, a schema of theorems.):Theorem 1.1. There is no nontrivial definable elementary embedding j: V → V.Many symbols and their definitions follow those used by Drake's book [1]. The formula Sat expresses the satisfaction relation . The formula Fmla(u) expresses that u is the Gödel-set for a formula.

Nonconstructive Properties of Well-Ordered T2 topological Spaces

We show that none of the following statements is provable in Zermelo-Fraenkel set theory (ZF) answering the corresponding open questions from Brunner in ``The axiom of choice in topology'': (i) For every T2 topological space (X, T) if X is well-ordered, then X has a well-ordered base, (ii) For every T2 topological space (X, T), if X is well-ordered, then there exists a function f : X × W $ \rightarrow$ T such that W is a well-ordered set and f ({x} × W) is a neighborhood base at x for each x $ \in$ X, (iii) For every T2 topological space (X, T), if X has a well-ordered dense subset, then there exists a function f : X × W $ \rightarrow$ T such that W is a well-ordered set and {x} = $ \cap$ f ({x} × W) for each x $ \in$ X.

Final coalgebras as greatest fixed points in ZF set theory

A special final coalgebra theorem, in the style of Aczel (1988), is proved within standard Zermelo–Fraenkel set theory. Aczel's Anti-Foundation Axiom is replaced by a variant definition of function that admits non-well-founded constructions. Variant ordered pairs and tuples, of possibly infinite length, are special cases of variant functions. Analogues of Aczel's solution and substitution lemmas are proved in the style of Rutten and Turi (1993). The approach is less general than Aczel's, but the treatment of non-well-founded objects is simple and concrete. The final coalgebra of a functor is its greatest fixedpoint.Compared with previous work (Paulson, 1995a), iterated substitutions and solutions are considered, as well as final coalgebras defined with respect to parameters. The disjoint sum construction is replaced by a smoother treatment of urelements that simplifies many of the derivations.The theory facilitates machine implementation of recursive definitions by letting both inductive and coinductive definitions be represented as fixed points. It has already been applied to the theorem prover Isabelle (Paulson, 1994).

Models of Second-Order Zermelo Set Theory

In [12], Ernst Zermelo described a succession of models for the axioms of set theory as initial segments of a cumulative hierarchy of levelsUαVα. The recursive definition of theVα's is:Thus, a little reflection on the axioms of Zermelo-Fraenkel set theory (ZF) shows thatVω, the first transfinite level of the hierarchy, is a model of all the axioms ofZFwith the exception of the axiom of infinity. And, in general, one finds that ifκis a strongly inaccessible ordinal, thenVκis a model of all of the axioms ofZF. (For all these models, we take∈to be the standard element-set relation restricted to the members of the domain.) Doubtless, when cast as a first-order theory,ZFdoes not characterize the structures 〈Vκ,∈∩(Vκ×Vκ)〉 forκa strongly inaccessible ordinal, by the Löwenheim-Skolem theorem. Still, one of the main achievements of [12] consisted in establishing that a characterization of these models can be attained when one ventures into second-order logic. For let second-orderZFbe, as usual, the theory that results fromZFwhen the axiom schema of replacement is replaced by its second-order universal closure. Then, it is a remarkable result due to Zermelo that second-orderZFcan only be satisfied in models of the form 〈Vκ,∈∩(Vκ×Vκ)〉 forκa strongly inaccessible ordinal.

Extension of standard models of ZFC to models of Nelson’s nonstandard set theory IST

A characterization of the standard models of ZFC set theory that are embeddable as the class of standard sets in models of the internal set theory IST and some of its versions is proposed.

Axiomatic derivation of the continuum hypothesis

After defining the Axiom of Monotonicity, it is used along with Zermelo-Fraenkel set theory to derive Cantor's Continuum Hypothesis. Several related theorems are also proved.

Toward quantum mathematics. I. From quantum set theory to universal quantum mechanics

We develop the old idea of von Neumann of a set theory with an internal quantum logic in a modern categorical guise [i.e., taking the objects of the category H of (pre-)Hilbert spaces and linear maps as the sets of the basic level]. We will see that in this way it is possible to clarify the relationship between categorification and quantization and besides this to understand that in some sense a categorificational approach to quantization is a discretized version of the one taken by noncommutative geometry. The tower of higher categorifications will appear as the analog of the von Neumann hierarchy of classical set theory (where by classical set theory, we will understand the usual Zermelo–Fraenkel system). Finally, we make a suggestion of how to understand all the different categorifications as different realizations of one and the same abstract structure by viewing quantum mechanics as universal in the sense of category theory. This gives the possibility to view extended topological quantum field theories purely as involving an abstract notion of quantum mechanics plus representation theory without the need to enlarge the class of kinematic structures of quantum systems on each step of categorification. In a future part of the work we will apply the language developed here to deal especially with the question of a categorification of the manifold notion.

The Hahn-Banach Property and the Axiom of Choice

We work in set theory ZF without axiom of choice. Though the Hahn-Banach theorem cannot be proved in ZF, we prove that every Gateaux-differentiable uniformly convex Banach space E satisfies the following continuous Hahn-Banach property: if p is a continuous sublinear functional on E, if F is a subspace of E, and if f: F ℝ is a linear functional such that f ≤ p|F then there exists a linear functional g : E ℝ such that g extends f and g ≤ p. We also prove that the continuous Hahn-Banach property on a topological vector space E is equivalent to the classical geometrical forms of the Hahn-Banach theorem on E. We then prove that the axiom of Dependent choices DC is equivalent to Ekeland's variational principle, and that it implies the continuous Hahn-Banach property on Gateaux-differentiable Banach spaces. Finally, we prove that, though separable normed spaces satisfy the continuous Hahn-Banach property, they do not satisfy the whole Hahn-Banach property in ZF+DC.

Finite Functions and the Necessary Use of Large Cardinals

Finite Functions and the Necessary Use of Large Cardinals

Countable choice and pseudometric spaces

In the realm of pseudometric spaces the role of choice principles is investigated. In particular it is shown that in ZF (i.e., Zermelo-Fraenkel set theory without the axiom of choice) the axiom of countable choice is not only sufficient but also necessary to establish each of the following results: 1. 1. separable ↔ countable base, 2. 2. separable ↔ Lindelöf, 3. 3. separable ↔ topologically totally bounded, 4. 4. compact → separable, 5. 5. separability is hereditary, 6. 6. the Baire Category Theorem for complete spaces with countable base, 7. 7. the Baire Category Theorem for complete, totally bounded spaces, 8. 8. compact ↔ sequentially compact, 9. 9. compact ↔ (totally bounded and complete), 10. 10. sequentially compact ↔ (totally bounded and complete), 11. 11. Weierstraβ compact ↔ (totally bounded and complete).