A system of axiomatic set theory - Part VII

Journal of Symbolic Logic 19 (2):81-96 (1954)
  Copy   BIBTEX

Abstract

The reader of Part VI will have noticed that among the set-theoretic models considered there some models were missing which were announced in Part II for certain proofs of independence. These models will be supplied now.Mainly two models have to be constructed: one with the property that there exists a set which is its own only element, and another in which the axioms I–III and VII, but not Va, are satisfied. In either case we need not satisfy the axiom of infinity. Thereby it becomes possible to set up the models on the basis of only I–III, and either VII or Va, a basis from which number theory can be obtained as we saw in Part II.On both these bases the Π0-system of Part VI, which satisfies the axioms I–V and VII, but not VI, can be constructed, as we stated there. An isomorphic model can also be obtained on that basis, by first setting up number theory as in Part II, and then proceeding as Ackermann did.Let us recall the main points of this procedure.For the sake of clarity in the discussion of this and the subsequent models, it will be necessary to distinguish precisely between the concepts which are relative to the basic set-theoretic system, and those which are relative to the model to be defined.

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 90,616

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Introduction to axiomatic set theory.Gaisi Takeuti - 1971 - New York,: Springer Verlag. Edited by Wilson M. Zaring.
A system of axiomatic set theory: Part IV. general set theory.Paul Bernays - 1942 - Journal of Symbolic Logic 7 (4):133-145.
A system of axiomatic set theory—Part II.Paul Bernays - 1941 - Journal of Symbolic Logic 6 (1):1-17.
A system of axiomatic set theory—Part VI.Paul Bernays - 1948 - Journal of Symbolic Logic 13 (2):65-79.

Analytics

Added to PP
2009-01-28

Downloads
53 (#268,501)

6 months
3 (#447,120)

Historical graph of downloads
How can I increase my downloads?

Citations of this work

Incompleteness Via Paradox and Completeness.Walter Dean - 2020 - Review of Symbolic Logic 13 (3):541-592.
The empty set, the Singleton, and the ordered pair.Akihiro Kanamori - 2003 - Bulletin of Symbolic Logic 9 (3):273-298.
Models of second-order zermelo set theory.Gabriel Uzquiano - 1999 - Bulletin of Symbolic Logic 5 (3):289-302.
The mathematical import of zermelo's well-ordering theorem.Akihiro Kanamori - 1997 - Bulletin of Symbolic Logic 3 (3):281-311.

View all 15 citations / Add more citations

References found in this work

No references found.

Add more references