Bi-Modal Naive Set Theory

This paper describes a modal conception of sets, according to which sets are 'potential' with respect to their members. A modal theory is developed, which invokes a naive comprehension axiom schema, modified by adding `forward looking' and `backward looking' modal operators. We show that this `bi-modal' naive set theory can prove modalized interpretations of several ZFC axioms, including the axiom of infinity. We also show that the theory is consistent by providing an S5 Kripke model. The paper concludes with some discussion of the nature of the modalities involved, drawing comparisons with noneism, the view that there are some non-existent objects.

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Predicate calculus and naive set theory in pure combinatory logic

Archive for Mathematical Logic ◽

10.1007/bf02011642 ◽

1981 ◽

Vol 21 (1) ◽

pp. 169-177 ◽

Cited By ~ 3

Author(s):

M. W. Bunder

Keyword(s):

Set Theory ◽

Predicate Calculus ◽

Combinatory Logic ◽

Naive Set Theory

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Naive Set Theory

Set Theory - Universitext ◽

10.1007/978-3-319-06725-4_1 ◽

2014 ◽

pp. 1-8

Author(s):

Ralf Schindler

Keyword(s):

Set Theory ◽

Naive Set Theory

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Naïve set theory

Springer Undergraduate Mathematics Series - Sets, Logic and Categories ◽

10.1007/978-1-4471-0589-3_1 ◽

1998 ◽

pp. 1-36

Author(s):

Peter J. Cameron

Keyword(s):

Set Theory ◽

Naive Set Theory

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Semantics for Naive Set Theory in Many-Valued Logics

The Age of Alternative Logics ◽

10.1007/978-1-4020-5012-7_9 ◽

2006 ◽

pp. 121-136

Author(s):

Thierry Libert

Keyword(s):

Set Theory ◽

Naive Set Theory

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Naive Set Theory

IFSR International Series on Systems Science and Engineering - General Systems Theory ◽

10.1007/0-306-46962-6_2 ◽

2006 ◽

pp. 15-57

Keyword(s):

Set Theory ◽

Naive Set Theory

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A method of modelling the formalism of set theory in axiomatic set theory

Journal of Symbolic Logic ◽

10.2307/2271333 ◽

1963 ◽

Vol 28 (1) ◽

pp. 20-34 ◽

Cited By ~ 4

Author(s):

A. H. Kruse

Keyword(s):

Set Theory ◽

Object Theory ◽

Axiom Schema ◽

Axiomatic Set Theory ◽

Adequate Analysis

As is well known, some paradoxes arise through inadequate analysis of the meanings of terms in a language, an adequate analysis showing that the paradoxes arise through a lack of separation of an object theory and a metatheory. Under such an adequate analysis in which parts of the metatheory are modelled in the object theory, the paradoxes give way to remarkable theorems establishing limitations of the object theory.Such a modelling is often accomplished by a Gödel numbering. Here we shall use a somewhat different technique in axiomatic set theory, from which we shall reap a few results having the effect of comparing the strength of various axiom schema of comprehension for sets and classes (cf. the numbered results of §§5–7). Similar results were obtained by A. Mostowski [7] using Gödel numbering (cf. 5.3 and 7.3 below).

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Global quantification in Zermelo-Fraenkel set theory

Journal of Symbolic Logic ◽

10.2307/2274215 ◽

1985 ◽

Vol 50 (2) ◽

pp. 289-301

Author(s):

John Mayberry

Keyword(s):

Set Theory ◽

Order Theory ◽

Prima Facie ◽

Local Theory ◽

Axiom Schema ◽

First Order ◽

First Order Theory ◽

Naturally Occurring ◽

Universe Of Sets

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.

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