Axiomatic Set Theory Research Articles

Azriel Lévy. Definability in axiomatic set theory I. Logic, methodology and philosophy of science, Proceedings of the 1964 International Congress, edited by Yehoshua Bar-Hillel, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam 1965, pp. 127–151.

Azriel Lévy. Definability in axiomatic set theory I. Logic, methodology and philosophy of science, Proceedings of the 1964 International Congress, edited by Yehoshua Bar-Hillel, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam 1965, pp. 127–151.

Introduction to Axiomatic Set Theory.

Journal Article Book Reviews Get access Introduction to Axiomatic Set Theory. By E. J. Lemmon. (London : Routledge & Kegan Paul. 1968. Pp. vi + 131. Price cloth 21s, paper 14s.) Alan Slomson Alan Slomson Search for other works by this author on: Oxford Academic Google Scholar The Philosophical Quarterly, Volume 20, Issue 78, January 1970, Pages 82–83, https://doi.org/10.2307/2217920 Published: 01 January 1970

Natural models of ackermann's set theory

In 1956 W. Ackermann proposed a new axiomatic set theory, that has received some attention in recent years. (See references [3], [4], [6], [7], and [9].) This theory distinguishes between sets and classes. In this paper we study mainly the natural models of this theory. We show, among other results, that the set-theoretical fragment of these models are also models of Zermelo-Fraenkel set theory. This result gives a partial answer to the question, raised by A. Levy, of the relative strength of Ackermann's set theory with respect of Zermelo-Fraenkel's.2

The effectivity of existential statements in axiomatic set theory

Mathematical experience indicates a close connection between the axiom of choice and noneffective existence proofs. In order to make this connection a subject of a systematic study it seems appropriate to approach this problem from the point of view of the hierarchy developed by the author in [8]. It turns out, as can be seen from various related results and examples, that the connection between the axiom of choice and effectivity comes up when one considers sentences of the form (∀ x)(∃ y) χ( x, y), where, throughout this abstract, χ( x, y) is a formula which does not contain quantifiers other than quantifiers restricted to sets (such as (∀ t)( t ϵ s → …) or (∃ t)( t ϵ s …). Many statements of set theory are of this form; e.g. the ordering theorem, the Boolean prime ideal theorem, the generalized continuum hypothesis. If one can prove in the Zermelo-Fraenkel set theory without the axiom of choice ( ZF) that (∀ x)(∃ y) χ( x, y) then there is a term τ( u) of set theory such that ZF ⥼(∀x) (there is a finite subset u of the transitive closure of x) χ( x, τ( u)). One cannot expect to get a “more effective” solution for y since no such solution exists even for y in (∀ x)(∃ y)( x = 0 v y ϵ x). If one can prove (∀ x)(∃ y) χ( x, y) with the aid of the axiom of choice then there is a term τ such that ZF ⥼(∀x) (for every well-ordering r of the transitive closure of x) χ( x, τ( r)).

On sets of almost disjoint subsets of a set

disjoint denumerable sets. In view of recent axiomatic results 4Pis independent of the usual axioms of set theory and the generalized continuumhypothesis. In § 4 we show that YE implies a certain unsolved problem of [3].In §5 we consider another question about sets of almost disjoint subsets of aset which was raised by F.

A. H. Kruse. A method of modelling the formalism of set theory in axiomatic set theory. The journal of symbolic logic, vol. 28 no. 1 (for 1963, pub. 1964), pp. 20–34.

A. H. Kruse. A method of modelling the formalism of set theory in axiomatic set theory. The journal of symbolic logic, vol. 28 no. 1 (for 1963, pub. 1964), pp. 20–34.

On a Theory Objects Based on a Single Axiom Scheme

There are some fundamental mathematical theories, such as the Fraenkel set-theory and the Bernays-Gödel set-theory, in which, I believe, all the actually important formal theories of mathematics can be embedded. Formal theories come into existence by being shown their consistency. As far as this is admitted, not all the axioms of set theory are necessary for a fundamental mathematical theory. The fundierung axiom is proved consistent by v. Neumann, the axiom of extensionality is proved consistent by Gandy, and even the axiom of choice is proved consistent by Göldel. Although it is not evident that a set-theory does not cease from being a fundamental theory of mathematics after abandoning these axioms all at once, the theory must be enough for being a fundamental theory of mathematics without some of them.

A minimal model for set theory

In the proof of the consistency of the Continuum Hypothesis and the Axiom of Choice with the other axioms of set theory, Godel [ l ] introduced the notion of a constructible set and showed that the constructible sets form a model for set theory. These sets are intuitively those which can be reached by means of a transfinite sequence of several simple operations. He then showed that the Axiom of Choice and Continuum Hypothesis held in the collection of constructible sets. If the original universe of sets is sufficiently rich in ordinal numbers, it will follow that every set is constructible, in which case we say that the Axiom of Constructibility is satisfied. This axiom implies the two axioms previously mentioned. However, from one point of view it may seem that this notion of constructibility does not intuitively correspond to what is meant by constructive since it may happen that all sets in the universe are constructive. In this paper we show that a more restricted notion of construction will yield a class of sets which form a minimal model for set theory. In this manner we prove the consistency of a stronger form of the Axiom of Constructibility. We observe that the idea of a minimal collection of objects satisfying certain axioms is well known in mathematics, for example, in group theory one often considers the subgroup generated by a collection of elements, and in measure theory we define the Borel sets as the smallest y) holds, then there exists a set B consisting of precisely those y. Since much of the proofs of the theorems we state follow quite closely the arguments of [ l ] , we shall be rather brief. Our main result is

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

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).

A contribution to Gödel's axiomatic set theory, III

A contribution to Gödel's axiomatic set theory, III

The Independence of a Strong Axiom of Choice

There is a growing realization among mathematicians and logicians of the many-sided role played by the axiom of choice in various branches of mathematics. Many of them tend to accept the axiom of choice as a legitimate principle provided, of course, it is proved to be independent in a suitable axiom system. This tendency has been accelerated by Gödel’s proof of the compatibility of this axiom in a reasonably broad system of axioms [2]. Such a view seems to have been shared by Fraenkel and Bar-Hillel [1; pp. 44-80] in their excellent exposition of the function of the axiom of choice in the modern mathematics in general and the axiomatic set theory in particular.

Axiomatic Set Theory.

Axiomatic Set Theory.

Azriel Lévy. Axiom schemata of strong infinity in axiomatic set theory. Pacific journal of mathematics, vol. 10 (1960), pp. 223–238.

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Patrick Suppes. Axiomatic set theory. Princeton: Van Nostrand Co., 1960. xii + 265 pp. $6.00.

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Axiomatic Set Theory.

Axiomatic Set Theory.

Axiomatic Set Theory.

Axiomatic Set Theory.

Book Review: Axiomatic set theory

Book Review: Axiomatic set theory

Axiom schemata of strong infinity in axiomatic set theory

Axiom schemata of strong infinity in axiomatic set theory

Note on axiomatic set theory. I. The independence of Zermelo's ``Aussonderungsaxiom'' from other axioms of set theory

Note on axiomatic set theory. I. The independence of Zermelo's ``Aussonderungsaxiom'' from other axioms of set theory

Principles of reflection in axiomatic set theory

Principles of reflection in axiomatic set theory