Zermelo-Fraenkel Research Articles

The Finite and the Infinite

In ZF set theory finiteness classes are introduced and their stability under basic set theoretical constructions are being investigated. Typical results are: 1. The class of finite sets is the smallest finiteness class. 2. The class of Dedekind-finite sets is the largest finiteness class. 3. The class of almost finite sets is the largest summable finiteness class. 4. Equivalent are: a. There is only one finiteness class. b. The union of each family of 1-element sets, indexed by a Dedekind-finite set, is Dedekind-finite. c. The axiom of choice, for countable families of Dedekind-finite sets. d. The shrinking principle for families (Xi)i ∈ I of sets, indexed by a Dedekind-finite set I (i.e., there exists a family (Yi)i ∈ I of pairwise disjoint subsets Yi of Xi with \(\underset{i\in I}{\bigcup} Y_i = \underset{i\in I}{\bigcup} X_i\)). 5. In suitable ZF-models there exist families \(({\mathfrak{A}}_r)_{r\in{\mathbb{R}}}\) of finiteness classes such that $$r < s \Longrightarrow {\mathfrak{A}}_r \subsetneqq {\mathfrak{A}}_s.$$

Three Models to Measure Information Security Compliance

This work introduces three models to measure information security compliance. These are the cardinality model, the second’s model, which is based on vector space, and the last model is based on the priority principle. Each of these models will be presented with definitions, basic operations, and examples. All three models are based on a new theory to understand information security called the Information Security Sets Theory (ISST). The ISST is based on four basic sets: external sets, local strategy sets, local standard sets, and local implementation sets. It should be noted that two sets are used to create local standard sets—local expansion and local creation. The major differences between the Zermelo Fraenkel set theory and the ISST are the elimination of using empty element and empty set. This assumption is based on “there is not empty security” measure and the is substituted to be and is defined as “minimum security (or system default security)”. The main objective of this article is to achieve new modeling system for information security compliance. The compliance measurement is defined in the first model as the cardinality between local strategy sets and the actual local implementation. The second model is looking at the security compliance as the angle between two sets, local implementation and local standard. The third model is based on the priority philosophy for local security standard.

Simplicity via Provability for Universal Prefix-free Turing Machines

Universality, provability and simplicity are key notions in computability theory. There are various criteria of simplicity for universal Turing machines. Probably the most popular one is to count the number of states/symbols. This criterion is more complex than it may appear at a first glance. In this note we propose three new criteria of simplicity for universal prefix-free Turing machines. These criteria refer to the possibility of proving various natural properties of such a machine (its universality, for example) in a formal theory, Peano arithmetic or Zermelo-Fraenkel set theory. In all cases some, but not all, machines are simple.

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.

Topological sums and products in ZF-set theory

In ZF, i.e., Zermelo–Fraenkel set theory without the axiom of choice, the category Top of topological spaces and continuous maps is well-behaved. In particular, Top has sums ( = coproducts ) and products. However, it may happen that for families ( X i ) i ∈ I and ( Y i ) i ∈ I with the property that each X i is homeomorphic to the corresponding Y i neither their sums ⊕ i ∈ I X i and ⊕ i ∈ I Y i nor their products ∏ i ∈ I X i and ∏ i ∈ I Y i are homeomorphic. It will be shown that the axiom of choice is not only sufficient but also necessary to rectify this defect.

On Interpretations of Bounded Arithmetic and Bounded Set Theory

In On interpretations of arithmetic and set theory, Kaye and Wong proved the following result, which they considered to belong to the folklore of mathematical logic. Theorem The first-order theories of Peano arithmetic and Zermelo-Fraenkel set theory with the axiom of infinity negated are bi-interpretable. In this note, I describe a theory of sets that is bi-interpretable with the theory of bounded arithmetic I Δ0 +exp . Because of the weakness of this theory of sets, I cannot straightforwardly adapt Kaye and Wong's interpretation of the arithmetic in the set theory. Instead, I am forced to produce a different interpretation.

Different versions of a first countable space without choice

We show that two versions of a first countable topological space which are equivalent in ZFC set theory split in the absence of the Axiom of Choice AC. This answers in the negative a related question from Gutierres “What is a first countable space?”.

Canonical subbase-compactness of topological products

For topological products the concept of canonical subbase-compactness is introduced, and the question analyzed under what conditions such products are canonically subbase-compact in ZF-set theory. Results: (1) Products of finite spaces are canonically subbase-compact iff AC ( fin ) , the axiom of choice for finite sets, holds. (2) Products of n-element spaces are canonically subbase-compact iff AC ( < n ) , the axiom of choice for sets with less than n elements, holds. (3) Products of compact spaces are canonically subbase-compact iff AC, the axiom of choice, holds. (4) All powers X I of a compact space X are canonically subbase compact iff X is a Loeb-space. These results imply that in ZF the implications compact ⇒ canonically subbase-compact ⇒ subbase-compact are both irreversible.

General relativistic hypercomputing and foundation of mathematics

Looking at very recent developments in spacetime theory, we can wonder whether these results exhibit features of hypercomputation that traditionally seemed impossible or absurd. Namely, we describe a physical device in relativistic spacetime which can compute a non-Turing computable task, e.g. which can decide the halting problem of Turing machines or decide whether ZF set theory is consistent (more precisely, can decide the theorems of ZF). Starting from this, we will discuss the impact of recent breakthrough results of relativity theory, black hole physics and cosmology to well established foundational issues of computability theory as well as to logic. We find that the unexpected, revolutionary results in the mentioned branches of science force us to reconsider the status of the physical Church Thesis and to consider it as being seriously challenged. We will outline the consequences of all this for the foundation of mathematics (e.g. to Hilbert's programme). Observational, empirical evidence will be quoted to show that the statements above do not require any assumption of some physical universe outside of our own one: in our specific physical universe there seem to exist regions of spacetime supporting potential non-Turing computations. Additionally, new "engineering" ideas will be outlined for solving the so-called blue-shift problem of GR-computing. Connections with related talks at the Physics and Computation meeting, e.g. those of Jerome Durand-Lose, Mark Hogarth and Martin Ziegler, will be indicated.

Where Our Number Concepts Come From

Where Our Number Concepts Come From

FREGE MEETS ZERMELO: A PERSPECTIVE ON INEFFABILITY AND REFLECTION

1. Philosophical background: iteration, ineffability, reflection. There are at least two heuristic motivations for the axioms of standard set theory, by which we mean, as usual, first-order Zermelo–Fraenkel set theory with the axiom of choice (ZFC): the iterative conception and limitation of size (see Boolos, 1989). Each strand provides a rather hospitable environment for the hypothesis that the set-theoretic universe is ineffable, which is our target in this paper, although the motivation is different in each case.

Model theory of the regularity and reflection schemes

This paper develops the model theory of ordered structures that satisfy Keisler’s regularity scheme and its strengthening REF $${(\mathcal{L})}$$ (the reflection scheme) which is an analogue of the reflection principle of Zermelo-Fraenkel set theory. Here $${\mathcal{L}}$$ is a language with a distinguished linear order <, and REF $${(\mathcal {L})}$$ consists of formulas of the form $$\exists x \forall y_{1} < x \ldots \forall y_{n} < x \varphi (y_{1},\ldots ,y_{n})\leftrightarrow \varphi^{ < x}(y_1, \ldots ,y_n),$$ where φ is an $${\mathcal{L}}$$ -formula, φ <x is the $${\mathcal{L}}$$ -formula obtained by restricting all the quantifiers of φ to the initial segment determined by x, and x is a variable that does not appear in φ. Our results include: The following five conditions are equivalent for a complete first order theory T in a countable language $${\mathcal{L}}$$ with a distinguished linear order: Moreover, if κ is a regular cardinal satisfying κ = κ <κ , then each of the above conditions is equivalent to:

Inconsistency of uncountable infinite sets under ZFC framework

PurposeThe purpose is to show that all uncountable infinite sets are self‐contradictory non‐sets.Design/methodology/approachA conceptual approach is taken in the paper.FindingsGiven the fact that the set N={x|n(x)} of all natural numbers, where n(x)=df “x is a natural number” is a self‐contradicting non‐set in this paper, the authors prove that in the framework of modern axiomatic set theory ZFC, various uncountable infinite sets are either non‐existent or self‐contradicting non‐sets. Therefore, it can be astonishingly concluded that in both the naive set theory or the modern axiomatic set theory, if any of the actual infinite sets exists, it must be a self‐contradicting non‐set.Originality/valueThe first time in history, it is shown that such convenient notion as the set of all real numbers needs to be reconsidered.

On the constructive Dedekind reals

In order to build the collection of Cauchy reals as a set in constructive set theory, the only power set-like principle needed is exponentiation. In contrast, the proof that the Dedekind reals form a set has seemed to require more than that. The main purpose here is to show that exponentiation alone does not suffice for the latter, by furnishing a Kripke model of constructive set theory, Constructive Zermelo–Fraenkel set theory with subset collection replaced by exponentiation, in which the Cauchy reals form a set while the Dedekind reals constitute a proper class.

The Relation Reflection Scheme

AbstractWe introduce a new axiom scheme for constructive set theory, the Relation Reflection Scheme (RRS). Each instance of this scheme is a theorem of the classical set theory ZF. In the constructive set theory CZF–, when the axiom scheme is combined with the axiom of Dependent Choices (DC), the result is equivalent to the scheme of Relative Dependent Choices (RDC). In contrast to RDC, the scheme RRS is preserved in Heyting‐valued models of CZF– using set‐generated frames. We give an application of the scheme to coinductive definitions of classes. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

On Interpretations of Arithmetic and Set Theory

This paper starts by investigating Ackermann's interpretation of finite set theory in the natural numbers. We give a formal version of this interpretation from Peano arithmetic (PA) to Zermelo-Fraenkel set theory with the infinity axiom negated (ZF−inf) and provide an inverse interpretation going the other way. In particular, we emphasize the precise axiomatization of our set theory that is required and point out the necessity of the axiom of transitive containment or (equivalently) the axiom scheme of ∈-induction. This clarifies the nature of the equivalence of PA and ZF−inf and corrects some errors in the literature. We also survey the restrictions of the Ackermann interpretation and its inverse to subsystems of PA and ZF−inf, where full induction, replacement, or separation is not assumed. The paper concludes with a discussion on the problems one faces when the totality of exponentiation fails, or when the existence of unordered pairs or power sets is not guaranteed.

Relating First-Order Set Theories and Elementary Toposes

AbstractWe show how to interpret the language of first-order set theory in an elementary topos endowed with, as extra structure, a directed structural system of inclusions (dssi). As our main result, we obtain a complete axiomatization of the intuitionistic set theory validated by all such interpretations. Since every elementary topos is equivalent to one carrying a dssi, we thus obtain a first-order set theory whose associated categories of sets are exactly the elementary toposes. In addition, we show that the full axiom of Separation is validated whenever the dssi is superdirected. This gives a uniform explanation for the known facts that cocomplete and realizability toposes provide models for Intuitionistic Zermelo–Fraenkel set theory (IZF).

The shrinking principle and the axiom of choice

The axiom of choice is equivalent to the shrinking principle: every indexed cover of a set has a refinement with the same index set which is a partition. A simple and direct proof of this equivalence is given within an elementary fragment of constructive Zermelo–Fraenkel set theory. Variants of the shrinking principle are discussed, and it is related to a similar but different principle formulated by Vaught.

Formula-inaccessible cardinals and a characterization of all natural models of Zermelo-Fraenkel set theory

E. Zermelo (1930) and J. C. Sheperdson (1952) proved that a cumulative set is a standard model of von Neumann-Bernays-Gödel set theory if and only if for some inaccessible cardinal number . The problem of a canonical form for all natural models of ZF theory turned out to be more complicated. Since the notion of a model of ZF theory cannot be defined by a finite set of formulae, we introduce a new notion of (strongly) formula-inaccessible cardinal number using a schema of formulae and its relativization on the set , and prove a formula-analogue of the Zermelo-Sheperdson theorem giving a canonical form for all natural models of ZF theory.

Patch-generated Frames and Projectable Hulls

This article considers coherent frame homomorphisms h : L → M between coherent frames, which induce an isomorphism between the boolen frames of polars, with M projectable, and such that M is generated by h(L) and certain complemented elements of M. This abstracts the passage from a semiprime commutative ring with identity to its projectable hull. The frame theoretic setting is investigated thoroughly, first without any assumptions beyond the Zermelo–Fraenkel axioms of set theory, and, subsequently, assuming that algebraic frames are spatial. The culmination of this effort is the result that the spectrum of d-elements of M is obtained from that of L by refining the given hull–kernel topology to the patch topology. The second part of the article relates the projectable hull to the (von Neumann) regular hull, in a variety of contexts, including that of f-rings. For a uniformly complete f-algebra A, it is shown that the maximal l-ideals of A that are traces of real maximal ideals of the regular hull HA are precisely the almost P-points of the space of maximal l-ideals of A.