Relevant bibliographies by topics / Zermelo-Fraenkel set theory / Journal articles
To see the other types of publications on this topic, follow the link: Zermelo-Fraenkel set theory.

Journal articles on the topic 'Zermelo-Fraenkel set theory'

Author: Grafiati
Published: 4 June 2021
Last updated: 20 June 2025

Create a spot-on reference in APA, MLA, Chicago, Harvard, and other styles

Select a source type:

Consult the top 50 journal articles for your research on the topic 'Zermelo-Fraenkel set theory.'

Next to every source in the list of references, there is an 'Add to bibliography' button. Press on it, and we will generate automatically the bibliographic reference to the chosen work in the citation style you need: APA, MLA, Harvard, Chicago, Vancouver, etc.

You can also download the full text of the academic publication as pdf and read online its abstract whenever available in the metadata.

Browse journal articles on a wide variety of disciplines and organise your bibliography correctly.

1

Mayberry, John. "Global quantification in Zermelo-Fraenkel set theory." Journal of Symbolic Logic 50, no. 2 (1985): 289–301. http://dx.doi.org/10.2307/2274215.

Full text
Abstract:
My aim here is to investigate the role of global quantifiers—quantifiers ranging over the entire universe of sets—in the formalization of Zermelo-Fraenkel set theory. The use of such quantifiers in the formulas substituted into axiom schemata introduces, at least prima facie, a strong element of impredicativity into the thapry. The axiom schema of replacement provides an example of this. For each instance of that schema enlarges the very domain over which its own global quantifiers vary. The fundamental question at issue is this: How does the employment of these global quantifiers, and the choice of logical principles governing their use, affect the strengths of the axiom schemata in which they occur?I shall attack this question by comparing three quite different formalizations of the intuitive principles which constitute the Zermelo-Fraenkel system. The first of these, local Zermelo-Fraenkel set theory (LZF), is formalized without using global quantifiers. The second, global Zermelo-Fraenkel set theory (GZF), is the extension of the local theory obtained by introducing global quantifiers subject to intuitionistic logical laws, and taking the axiom schema of strong collection (Schema XII, §2) as an additional assumption of the theory. The third system is the conventional formalization of Zermelo-Fraenkel as a classical, first order theory. The local theory, LZF, is already very strong, indeed strong enough to formalize any naturally occurring mathematical argument. I have argued (in [3]) that it is the natural formalization of naive set theory. My intention, therefore, is to use it as a standard against which to measure the strength of each of the other two systems.
APA, Harvard, Vancouver, ISO, and other styles
2

Hinnion, R. "Extensionality in Zermelo-Fraenkel Set Theory." Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 32, no. 1-5 (1986): 51–60. http://dx.doi.org/10.1002/malq.19860320107.

Full text
APA, Harvard, Vancouver, ISO, and other styles
3

Herman, Ari, and John Caughman. "Probability Axioms and Set Theory Paradoxes." Symmetry 13, no. 2 (2021): 179. http://dx.doi.org/10.3390/sym13020179.

Full text
Abstract:
In this paper, we show that Zermelo–Fraenkel set theory with Choice (ZFC) conflicts with basic intuitions about randomness. Our background assumptions are the Zermelo–Fraenekel axioms without Choice (ZF) together with a fragment of Kolmogorov’s probability theory. Using these minimal assumptions, we prove that a weak form of Choice contradicts two common sense assumptions about probability—both based on simple notions of symmetry and independence.
APA, Harvard, Vancouver, ISO, and other styles
4

Enayat, Ali, and Mateusz Łełyk. "Categoricity-like Properties in the First Order Realm." Journal for the Philosophy of Mathematics 1 (September 10, 2024): 63–98. http://dx.doi.org/10.36253/jpm-2934.

Full text
Abstract:
By classical results of Dedekind and Zermelo, second order logic imposes categoricity features on Peano Arithmetic and Zermelo-Fraenkel set theory. However, we have known since Skolem’s anti-categoricity theorems that the first order formulations of Peano Arithmetic and Zermelo- Fraenkel set theory (i.e., PA and ZF) are not categorical. Here we investigate various categoricity-like properties (including tightness, solidity, and internal categoricity) that are exhibited by a distinguished class of first order theories that include PA and ZF, with the aim of understanding what is special about canonical foundational first order theories.
APA, Harvard, Vancouver, ISO, and other styles
5

ARAI, TOSHIYASU. "LIFTING PROOF THEORY TO THE COUNTABLE ORDINALS: ZERMELO-FRAENKEL SET THEORY." Journal of Symbolic Logic 79, no. 2 (2014): 325–54. http://dx.doi.org/10.1017/jsl.2014.6.

Full text
APA, Harvard, Vancouver, ISO, and other styles
6

Gabbay, Murdoch. "Equivariant ZFA and the foundations of nominal techniques." Journal of Logic and Computation 30, no. 2 (2020): 525–48. http://dx.doi.org/10.1093/logcom/exz015.

Full text
Abstract:
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.
APA, Harvard, Vancouver, ISO, and other styles
7

Chen, Ray-Ming, and Michael Rathjen. "Lifschitz realizability for intuitionistic Zermelo–Fraenkel set theory." Archive for Mathematical Logic 51, no. 7-8 (2012): 789–818. http://dx.doi.org/10.1007/s00153-012-0299-2.

Full text
APA, Harvard, Vancouver, ISO, and other styles
8

LÖWE, BENEDIKT, and SOURAV TARAFDER. "GENERALIZED ALGEBRA-VALUED MODELS OF SET THEORY." Review of Symbolic Logic 8, no. 1 (2015): 192–205. http://dx.doi.org/10.1017/s175502031400046x.

Full text
Abstract:
AbstractWe generalize the construction of lattice-valued models of set theory due to Takeuti, Titani, Kozawa and Ozawa to a wider class of algebras and show that this yields a model of a paraconsistent logic that validates all axioms of the negation-free fragment of Zermelo-Fraenkel set theory.
APA, Harvard, Vancouver, ISO, and other styles
9

Balogun, F., O. A. Wahab, and A. I. Isah. "Axiomatization multisets: a comparative analysis." Dutse Journal of Pure and Applied Sciences 9, no. 3b (2023): 155–63. http://dx.doi.org/10.4314/dujopas.v9i3b.17.

Full text
Abstract:
A multiset, unlike the classical set, allows for multiple instances of its elements. In this paper, we present a comparative analysis of theories on multisets. In particular, we examine, the first-order two-sorted multiset theory MST, and the single-sorted multiset theory MS that employs the same sort for multiplicities and the set they support. The logical strengths and significance of some axioms presented in these theories are investigated. The theory MST contains a copy of the Zermelo-Fraenkel set theory with the axiom of choice (ZFC) but is independent of ZFC. The single-sorted multiset theory describes a stronger theory that mirrors the Zermelo-Fraenkel set theory (ZF) and is equiconsistent with ZF and antifoundation. The two-sorted multisettheory MST is a conservative extension of the classical set theory, making it a suitable theory to assume when dealing with studies that involve multisets.
APA, Harvard, Vancouver, ISO, and other styles
10

Mardanov, Asliddin Khasamiddinovich. "SET THEORY: THE STUDY OF SETS, THEIR OPERATIONS, AND THE RELATIONS BETWEEN THEM." Multidisciplinary Journal of Science and Technology 5, no. 1 (2025): 622–23. https://doi.org/10.5281/zenodo.14816736.

Full text
Abstract:
Set theory is a fundamental branch of mathematics that deals with the study of sets, their operations, and the relationships between them. A set is defined as a collection of distinct objects, and set theory provides the formal framework for understanding how these collections interact. This article explores the foundational concepts of set theory, including set operations, relations, and their significance in mathematics. It also discusses key results in the theory, such as the Axiom of Choice, the Zermelo-Fraenkel axioms, and the concept of cardinality, as well as the role of set theory in other mathematical fields.
APA, Harvard, Vancouver, ISO, and other styles
11

Shirahata, Masaru. "A linear conservative extension of Zermelo-Fraenkel set theory." Studia Logica 56, no. 3 (1996): 361–92. http://dx.doi.org/10.1007/bf00372772.

Full text
APA, Harvard, Vancouver, ISO, and other styles
12

LINNEBO, ØYSTEIN. "THE POTENTIAL HIERARCHY OF SETS." Review of Symbolic Logic 6, no. 2 (2013): 205–28. http://dx.doi.org/10.1017/s1755020313000014.

Full text
Abstract:
AbstractSome reasons to regard the cumulative hierarchy of sets as potential rather than actual are discussed. Motivated by this, a modal set theory is developed which encapsulates this potentialist conception. The resulting theory is equi-interpretable with Zermelo Fraenkel set theory but sheds new light on the set-theoretic paradoxes and the foundations of set theory.
APA, Harvard, Vancouver, ISO, and other styles
13

Rathjen, Michael. "The disjunction and related properties for constructive Zermelo-Fraenkel set theory." Journal of Symbolic Logic 70, no. 4 (2005): 1233–54. http://dx.doi.org/10.2178/jsl/1129642124.

Full text
Abstract:
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.
APA, Harvard, Vancouver, ISO, and other styles
14

Windsteiger, Wolfgang. "An automated prover for Zermelo–Fraenkel set theory in Theorema." Journal of Symbolic Computation 41, no. 3-4 (2006): 435–70. http://dx.doi.org/10.1016/j.jsc.2005.04.013.

Full text
APA, Harvard, Vancouver, ISO, and other styles
15

Bunina, E. I., and V. K. Zakharov. "Canonical form of Tarski sets in Zermelo-Fraenkel set theory." Mathematical Notes 77, no. 3-4 (2005): 297–306. http://dx.doi.org/10.1007/s11006-005-0030-2.

Full text
APA, Harvard, Vancouver, ISO, and other styles
16

Enayat, Ali. "Leibnizian models of set theory." Journal of Symbolic Logic 69, no. 3 (2004): 775–89. http://dx.doi.org/10.2178/jsl/1096901766.

Full text
Abstract:
Abstract.A model is said to be Leibnizian if it has no pair of indiscernibles. Mycielski has shown that there is a first order axiom LM (the Leibniz-Mycielski axiom) such that for any completion T of Zermelo-Fraenkel set theory ZF. T has a Leibnizian model if and only if T proves LM. Here we prove:Theorem A. Every complete theory T extending ZF + LM has nonisomorphic countable Leibnizian models.Theorem B. If κ is a prescribed definable infinite cardinal ofa complete theory T extending ZF + V = OD, then there are nonisomorphic Leibnizian models of T of power ℵ1such thatis ℵ1-like.Theorem C. Every complete theory T extendingZF + V = ODhas nonisomorphic ℵ1-like Leibnizian models.
APA, Harvard, Vancouver, ISO, and other styles
17

Pallares Vega, Ivonne. "Sets, Properties and Truth Values: A Category-Theoretic Approach to Zermelo’s Axiom of Separation." Athens Journal of Philosophy 1, no. 3 (2022): 135–62. http://dx.doi.org/10.30958/ajphil.1-3-2.

Full text
Abstract:
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
APA, Harvard, Vancouver, ISO, and other styles
18

Uzquiano, Gabriel. "Models of Second-Order Zermelo Set Theory." Bulletin of Symbolic Logic 5, no. 3 (1999): 289–302. http://dx.doi.org/10.2307/421182.

Full text
Abstract:
In [12], Ernst Zermelo described a succession of models for the axioms of set theory as initial segments of a cumulative hierarchy of levels UαVα. The recursive definition of the Vα's is:Thus, a little reflection on the axioms of Zermelo-Fraenkel set theory (ZF) shows that Vω, the first transfinite level of the hierarchy, is a model of all the axioms of ZF with the exception of the axiom of infinity. And, in general, one finds that if κ is a strongly inaccessible ordinal, then Vκ is a model of all of the axioms of ZF. (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, ZF does not characterize the structures 〈Vκ,∈∩(Vκ×Vκ)〉 for κ a 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-order ZF be, as usual, the theory that results from ZF when the axiom schema of replacement is replaced by its second-order universal closure. Then, it is a remarkable result due to Zermelo that second-order ZF can only be satisfied in models of the form 〈Vκ,∈∩(Vκ×Vκ)〉 for κ a strongly inaccessible ordinal.
APA, Harvard, Vancouver, ISO, and other styles
19

Boulabiar, Karim. "Lattice and Algebra Homomorphisms onC(X) in Zermelo-Fraenkel Set Theory." Quaestiones Mathematicae 38, no. 6 (2015): 835–39. http://dx.doi.org/10.2989/16073606.2014.981741.

Full text
APA, Harvard, Vancouver, ISO, and other styles
20

Swan, Andrew W. "A class of higher inductive types in Zermelo‐Fraenkel set theory." Mathematical Logic Quarterly 68, no. 1 (2022): 118–27. http://dx.doi.org/10.1002/malq.202100040.

Full text
APA, Harvard, Vancouver, ISO, and other styles
21

Heidema, Johannes. "An axiom schema of comprehension of zermelo–fraenkel–skolem set theory." History and Philosophy of Logic 11, no. 1 (1990): 59–65. http://dx.doi.org/10.1080/01445349008837157.

Full text
APA, Harvard, Vancouver, ISO, and other styles
22

Rathjen, Michael. "Constructive Zermelo–Fraenkel set theory and the limited principle of omniscience." Annals of Pure and Applied Logic 165, no. 2 (2014): 563–72. http://dx.doi.org/10.1016/j.apal.2013.08.001.

Full text
APA, Harvard, Vancouver, ISO, and other styles
23

Dihoum, Eman, and Michael Rathjen. "Preservation of choice principles under realizability." Logic Journal of the IGPL 27, no. 5 (2019): 746–65. http://dx.doi.org/10.1093/jigpal/jzz002.

Full text
Abstract:
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).
APA, Harvard, Vancouver, ISO, and other styles
24

Howard, Paul. "Unions of well-ordered sets." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 56, no. 1 (1994): 117–24. http://dx.doi.org/10.1017/s1446788700034753.

Full text
Abstract:
AbstractIn Zermelo-Fraenkel set theory weakened to permit the existence of atoms and without the axiom of choice we investigate the deductive strength of five statements which make assertions about the cardinality of the union of a well-ordered collection of sets. All five of the statements considered are consequences of the axiom of choice.
APA, Harvard, Vancouver, ISO, and other styles
25

Farah, Ilijas, and Ilan Hirshberg. "Simple nuclear C*-algebras not isomorphic to their opposites." Proceedings of the National Academy of Sciences 114, no. 24 (2017): 6244–49. http://dx.doi.org/10.1073/pnas.1619936114.

Full text
Abstract:
We show that it is consistent with Zermelo–Fraenkel set theory with the axiom of choice (ZFC) that there is a simple nuclear nonseparable C∗-algebra, which is not isomorphic to its opposite algebra. We can furthermore guarantee that this example is an inductive limit of unital copies of the Cuntz algebra O2 or of the canonical anticommutation relations (CAR) algebra.
APA, Harvard, Vancouver, ISO, and other styles
26

ALEXANDRU, ANDREI, and GABRIEL CIOBANU. "Uniformly supported sets and fixed points properties." Carpathian Journal of Mathematics 36, no. 3 (2020): 351–64. http://dx.doi.org/10.37193/cjm.2020.03.03.

Full text
Abstract:
The theory of finitely supported algebraic structures is a reformulation of Zermelo-Fraenkel set theory in which every set-based construction is finitely supported according to a canonical 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. Particularly, we focus on fixed points properties.
APA, Harvard, Vancouver, ISO, and other styles
27

Bunina, E. I., and V. K. Zakharov. "A canonical form for supertransitive standard models in Zermelo-Fraenkel set theory." Russian Mathematical Surveys 58, no. 4 (2003): 782–83. http://dx.doi.org/10.1070/rm2003v058n04abeh000646.

Full text
APA, Harvard, Vancouver, ISO, and other styles
28

Kanovei and Katz. "A Positive Function with Vanishing Lebesgue Integral in Zermelo–Fraenkel Set Theory." Real Analysis Exchange 42, no. 2 (2017): 385. http://dx.doi.org/10.14321/realanalexch.42.2.0385.

Full text
APA, Harvard, Vancouver, ISO, and other styles
29

Lewis, Alain A. "On the independence of core-equivalence results from Zermelo-Fraenkel set theory." Mathematical Social Sciences 19, no. 1 (1990): 55–95. http://dx.doi.org/10.1016/0165-4896(90)90038-9.

Full text
APA, Harvard, Vancouver, ISO, and other styles
30

Rathjen, Michael. "Replacement versus collection and related topics in constructive Zermelo–Fraenkel set theory." Annals of Pure and Applied Logic 136, no. 1-2 (2005): 156–74. http://dx.doi.org/10.1016/j.apal.2005.05.010.

Full text
APA, Harvard, Vancouver, ISO, and other styles
31

Pincus, David. "The dense linear ordering principle." Journal of Symbolic Logic 62, no. 2 (1997): 438–56. http://dx.doi.org/10.2307/2275540.

Full text
Abstract:
AbstractLet DO denote the principle: Every infinite set has a dense linear ordering. DO is compared to other ordering principles such as O, the Linear Ordering principle, KW, the Kinna-Wagner Principle, and PI, the Prime Ideal Theorem, in ZF, Zermelo-Fraenkel set theory without AC, the Axiom of Choice.The main result is:Theorem. AC ⇒ KW ⇒ DO ⇒ O, and none of the implications is reversible in ZF + PI.The first and third implications and their irreversibilities were known. The middle one is new. Along the way other results of interest are established. O, while not quite implying DO, does imply that every set differs finitely from a densely ordered set. The independence result for ZF is reduced to one for Fraenkel-Mostowski models by showing that DO falls into two of the known classes of statements automatically transferable from Fraenkel-Mostowski to ZF models. Finally, the proof of PI in the Fraenkel-Mostowski model leads naturally to versions of the Ramsey and Ehrenfeucht-Mostowski theorems involving sets that are both ordered and colored.
APA, Harvard, Vancouver, ISO, and other styles
32

Awodey, Steve, Carsten Butz, Alex Simpson, and Thomas Streicher. "Relating First-Order Set Theories and Elementary Toposes." Bulletin of Symbolic Logic 13, no. 3 (2007): 340–58. http://dx.doi.org/10.2178/bsl/1186666150.

Full text
Abstract:
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).
APA, Harvard, Vancouver, ISO, and other styles
33

VÄÄNÄNEN, JOUKO. "AN EXTENSION OF A THEOREM OF ZERMELO." Bulletin of Symbolic Logic 25, no. 2 (2019): 208–12. http://dx.doi.org/10.1017/bsl.2019.15.

Full text
Abstract:
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].
APA, Harvard, Vancouver, ISO, and other styles
34

Van, Der Poll John Andrew Andrew, Paula Kotzé, and Willem Labuschagne. "Automated Support for Enterprise Information Systems." JUCS - Journal of Universal Computer Science 10, no. (11) (2004): 1519–39. https://doi.org/10.3217/jucs-010-11-1519.

Full text
Abstract:
A condensed specification of a multi-level marketing (MLM) enterprise which can be modelled by mathematical forests and trees is presented in Z. We thereafter identify a number of proof obligations that result from operations on the state space. Z is based on first-order logic and a strongly-typed fragment of Zermelo-Fraenkel set theory, hence the feasibility of using certain reasoning heuristics developed for proving theorems in set theory is investigated for discharging the identified proof obligations. Using the automated reasoner OTTER, we illustrate how these proof obligations from a real-life enterprise may successfully be discharged using a suite of well-chosen heuristics.
APA, Harvard, Vancouver, ISO, and other styles
35

Esser, Olivier. "An Interpretation of the Zermelo-Fraenkel Set Theory and the Kelley-Morse Set Theory in a Positive Theory." Mathematical Logic Quarterly 43, no. 3 (1997): 369–77. http://dx.doi.org/10.1002/malq.19970430309.

Full text
APA, Harvard, Vancouver, ISO, and other styles
36

Zieliński, Marcin. "Cardinality of the sets of all bijections, injections and surjections." Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia 11 (February 5, 2020): 143–51. http://dx.doi.org/10.24917/20809751.11.8.

Full text
Abstract:
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.
APA, Harvard, Vancouver, ISO, and other styles
37

DEISER, OLIVER. "AN AXIOMATIC THEORY OF WELL-ORDERINGS." Review of Symbolic Logic 4, no. 2 (2011): 186–204. http://dx.doi.org/10.1017/s1755020310000390.

Full text
Abstract:
We introduce a new simple first-order framework for theories whose objects are well-orderings (lists). A system ALT (axiomatic list theory) is presented and shown to be equiconsistent with ZFC (Zermelo Fraenkel Set Theory with the Axiom of Choice). The theory sheds new light on the power set axiom and on Gödel’s axiom of constructibility. In list theory there are strong arguments favoring Gödel’s axiom, while a bare analogon of the set theoretic power set axiom looks artificial. In fact, there is a natural and attractive modification of ALT where every object is constructible and countable. In order to substantiate our foundational interest in lists, we also compare sets and lists from the perspective of finite objects, arguing that lists are, from a certain point of view, conceptually simpler than sets.
APA, Harvard, Vancouver, ISO, and other styles
38

Burns, R. G., John Lawrence, and Frank Okoh. "On the number of normal subgroups of an uncountable group." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 41, no. 3 (1986): 343–51. http://dx.doi.org/10.1017/s1446788700033802.

Full text
Abstract:
AbstractIn this paper two theorems are proved that give a partial answer to a question posed by G. Behrendt and P. Neumann. Firstly, the existence of a group of cardinality ℵ1 with exactly ℵ1 normal subgroups, yet having a subgroup of index 2 with 2ℵ1 normal subgroups, is consistent with ZFC (the Zermelo-Fraenkel axioms for set theory together with the Axiom of Choice). Secondly, the statement “Every metabelian-by-finite group of cardinality ℵ1 has 2ℵ1 normal subgroups” is consistent with ZFC.
APA, Harvard, Vancouver, ISO, and other styles
39

ACZEL, PETER, HAJIME ISHIHARA, TAKAKO NEMOTO, and YASUSHI SANGU. "Generalized geometric theories and set-generated classes." Mathematical Structures in Computer Science 25, no. 7 (2014): 1466–83. http://dx.doi.org/10.1017/s0960129513000236.

Full text
Abstract:
We introduce infinitary propositional theories over a set and their models which are subsets of the set, and define a generalized geometric theory as an infinitary propositional theory of a special form. The main result is thatthe class of models of a generalized geometric theory is set-generated. Here, a class$\mathcal{X}$of subsets of a set is set-generated if there exists a subsetGof$\mathcal{X}$such that for each α ∈$\mathcal{X}$, and finitely enumerable subset τ of α there exists a subset β ∈Gsuch that τ ⊆ β ⊆ α. We show the main result in the constructive Zermelo–Fraenkel set theory (CZF) with an additional axiom, called the set generation axiom which is derivable inCZF, both from the relativized dependent choice scheme and from a regular extension axiom. We give some applications of the main result to algebra, topology and formal topology.
APA, Harvard, Vancouver, ISO, and other styles
40

Nescolarde-Selva, Josué-Antonio, José-Luis Usó-Doménech, Lorena Segura-Abad, Kristian Alonso-Stenberg, and Hugh Gash. "Solutions of Extension and Limits of Some Cantorian Paradoxes." Mathematics 8, no. 4 (2020): 486. http://dx.doi.org/10.3390/math8040486.

Full text
Abstract:
Cantor thought of the principles of set theory or intuitive principles as universal forms that can apply to any actual or possible totality. This is something, however, which need not be accepted if there are totalities which have a fundamental ontological value and do not conform to these principles. The difficulties involved are not related to ontological problems but with certain peculiar sets, including the set of all sets that are not members of themselves, the set of all sets, and the ordinal of all ordinals. These problematic totalities for intuitive theory can be treated satisfactorily with the Zermelo and Fraenkel (ZF) axioms or the von Neumann, Bernays, and Gödel (NBG) axioms, and the iterative conceptions expressed in them.
APA, Harvard, Vancouver, ISO, and other styles
41

Zheng, Yaming. "The continuum hypothesis: Its independence from Zermelo-Fraenkel set theory and impact on mathematical foundations." Theoretical and Natural Science 13, no. 1 (2023): 293–97. http://dx.doi.org/10.54254/2753-8818/13/20240865.

Full text
Abstract:
The Continuum Hypothesis, originally posited by the pioneering mathematician Georg Cantor in the latter part of the 19th century, stands as a cornerstone inquiry in the realm of set theory. This paper embarks on a journey, delving into the rudiments of set theory, before tracing the evolutionary trajectory of the Continuum Hypothesis. Central to this exploration is the Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) a foundational pillar in modern set theoretic studies. The core tenets of ZFC are dissected, shedding light on the seminal proofs presented by luminaries in the field that underline the unprovability of the CH within this axiomatic system. Beyond its mathematical intricacies, the paper underscores the profound philosophical and practical implications of the CH in both set theory and the broader mathematical landscape. In synthesizing these insights, a profound realization emerges: the inherent limitations in establishing the veracity of the Continuum Hypothesis within the confines of ZFC. This poignant revelation beckons deeper introspection into the foundational underpinnings of mathematics, stirring both intrigue and reflection amongst scholars and enthusiasts alike.
APA, Harvard, Vancouver, ISO, and other styles
42

Dalen, Dirk Van, and Heinz-Dieter Ebbinghaus. "Zermelo and the Skolem Paradox." Bulletin of Symbolic Logic 6, no. 2 (2000): 145–61. http://dx.doi.org/10.2307/421203.

Full text
Abstract:
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”.
APA, Harvard, Vancouver, ISO, and other styles
43

Sieg, Wilfried. "The Cantor–Bernstein theorem: how many proofs?" Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 377, no. 2140 (2019): 20180031. http://dx.doi.org/10.1098/rsta.2018.0031.

Full text
Abstract:
Dedekind's proof of the Cantor–Bernstein theorem is based on his chain theory, not on Cantor's well-ordering principle. A careful analysis of the proof extracts an argument structure that can be seen in the many other proofs that have been given since. I contend there is essentially one proof that comes in two variants due to Dedekind and Zermelo , respectively. This paper is a case study in analysing proofs of a single theorem within a given methodological framework, here Zermelo–Fraenkel set theory (ZF). It uses tools from proof theory, but focuses on heuristic ideas that shape proofs and on logical strategies that help to construct them. It is rooted in a perspective on Beweistheorie that predates its close connection and almost exclusive attention to the goals of Hilbert's finitist consistency programme. This earlier perspective can be brought to life (only) with the support of powerful computational tools. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.
APA, Harvard, Vancouver, ISO, and other styles
44

Schrittesser, David, and Asger Törnquist. "The Ramsey property implies no mad families." Proceedings of the National Academy of Sciences 116, no. 38 (2019): 18883–87. http://dx.doi.org/10.1073/pnas.1906183116.

Full text
Abstract:
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 E0 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.
APA, Harvard, Vancouver, ISO, and other styles
45

SONPANOW, NATTAPON, and PIMPEN VEJJAJIVA. "A FINITE-TO-ONE MAP FROM THE PERMUTATIONS ON A SET." Bulletin of the Australian Mathematical Society 95, no. 2 (2016): 177–82. http://dx.doi.org/10.1017/s0004972716000757.

Full text
Abstract:
Forster [‘Finite-to-one maps’, J. Symbolic Logic68 (2003), 1251–1253] showed, in Zermelo–Fraenkel set theory, that if there is a finite-to-one map from ${\mathcal{P}}(A)$, the set of all subsets of a set $A$, onto $A$, then $A$ must be finite. If we assume the axiom of choice (AC), the cardinalities of ${\mathcal{P}}(A)$ and the set $S(A)$ of permutations on $A$ are equal for any infinite set $A$. In the absence of AC, we cannot make any conclusion about the relationship between the two cardinalities for an arbitrary infinite set. In this paper, we give a condition that makes Forster’s theorem, with ${\mathcal{P}}(A)$ replaced by $S(A)$, provable without AC.
APA, Harvard, Vancouver, ISO, and other styles
46

Bunina, E. I., and V. K. Zakharov. "Formula-inaccessible cardinals and a characterization of all natural models of Zermelo-Fraenkel set theory." Izvestiya: Mathematics 71, no. 2 (2007): 219–45. http://dx.doi.org/10.1070/im2007v071n02abeh002356.

Full text
APA, Harvard, Vancouver, ISO, and other styles
47

Gregoriades, Vassilios. "A recursion theoretic characterization of the Topological Vaught Conjecture in the Zermelo-Fraenkel set theory." Mathematical Logic Quarterly 63, no. 6 (2017): 544–51. http://dx.doi.org/10.1002/malq.201600094.

Full text
APA, Harvard, Vancouver, ISO, and other styles
48

Farah, Ilijas, and Saharon Shelah. "RIGIDITY OF CONTINUOUS QUOTIENTS." Journal of the Institute of Mathematics of Jussieu 15, no. 1 (2014): 1–28. http://dx.doi.org/10.1017/s1474748014000218.

Full text
Abstract:
We study countable saturation of metric reduced products and introduce continuous fields of metric structures indexed by locally compact, separable, completely metrizable spaces. Saturation of the reduced product depends both on the underlying index space and the model. By using the Gelfand–Naimark duality we conclude that the assertion that the Stone–Čech remainder of the half-line has only trivial automorphisms is independent from ZFC (Zermelo-Fraenkel axiomatization of set theory with the Axiom of Choice). Consistency of this statement follows from the Proper Forcing Axiom, and this is the first known example of a connected space with this property.
APA, Harvard, Vancouver, ISO, and other styles
49

Asselmeyer-Maluga, Torsten, and Jerzy Król. "Local External/Internal Symmetry of Smooth Manifolds and Lack of Tovariance in Physics." Symmetry 11, no. 12 (2019): 1429. http://dx.doi.org/10.3390/sym11121429.

Full text
Abstract:
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.
APA, Harvard, Vancouver, ISO, and other styles
50

Alexandru, Andrei, and Gabriel Ciobanu. "Soft Sets with Atoms." Mathematics 10, no. 12 (2022): 1956. http://dx.doi.org/10.3390/math10121956.

Full text
Abstract:
The theory of finitely supported structures is used for dealing with very large sets having a certain degree of symmetry. This framework generalizes the classical set theory of Zermelo-Fraenkel by allowing infinitely many basic elements with no internal structure (atoms) and by equipping classical sets with group actions of the permutation group over these basic elements. On the other hand, soft sets represent a generalization of the fuzzy sets to deal with uncertainty in a parametric manner. In this paper, we study the soft sets in the new framework of finitely supported structures, associating to any crisp set a family of atoms describing it. We prove some finiteness properties for infinite soft sets, some order properties and Tarski-like fixed point results for mappings between soft sets with atoms.
APA, Harvard, Vancouver, ISO, and other styles
We offer discounts on all premium plans for authors whose works are included in thematic literature selections. Contact us to get a unique promo code!

To the bibliography