Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-25T20:47:01.414Z Has data issue: false hasContentIssue false
  • English
  • Français

New set-theoretic axioms derived from a lean metamathematics

Published online by Cambridge University Press:  12 March 2014

Jan Mycielski
Jan Mycielski*
Affiliation:
Department of Mathematics, University of Colorado, Boulder, Colorado 80309-0395
Get access Rights & Permissions [Opens in a new window]

Extract

I will formulate in this paper three set-theoretic axioms, (A1), (A2), and (A3), which appear natural and settle some well-known questions, and I will give some metamathematical evidence supporting these axioms. I build upon a distinction between pure mathematics and applied metamathematics which views the first as an art dealing with imaginary objects, where following Poincaré we can say to exist is to be free of contradiction, and the second as a science describing the phenomenon of mathematics, where science means a description of some physical reality (in this case the reality of thoughts in our brains which underlie spoken or written mathematics). Of course this distinction is not new, but it has been disregarded by many mathematicians and philosophers who wrote about the nature of mathematics, e.g. by the Platonists, and even by some empiricists who thought that mathematics is a science. Since applied metamathematics is a science, unlike pure mathematics it has to be lean, i.e., to obey Ockham's principle of economy of concepts (entia non sunt multiplicanda praeter necessitatem).

In this lean metamathematics we describe pure mathematics as a finite structure of thoughts in our brains, and we think that written or spoken mathematics is an abstract description of this structure. (I think that this is the view which Poincaré expressed informally in his discussions with the logicians and set theorists of his times, although his idea has to be modified to some extent; see Remark 4 at the end of this paper.) We claim that all mathematical objects which are imagined when we develop a (first-order) theory T can be represented by means of terms of the language of a Skolemization of the set of sentences consisting of the axioms of T and of all theorems of logic.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 60 , Issue 1 , March 1995 , pp. 191 - 198
Copyright
Copyright © Association for Symbolic Logic 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Martin, D. A. and Steel, J. R., A proof of projective determinacy, Journal of the American Mathematical Society, vol. 2 (1989), pp. 71–125.CrossRefGoogle Scholar
[2]Montague, R. and Vaught, R. L., Natural models of set theories, Fundamenta Mathematicae, vol. 47 (1959), pp. 219–242.CrossRefGoogle Scholar
[3]Mycielski, J., On the psychological mechanism inducing human convictions, (manuscript to be published).Google Scholar
[4]Mycielski, J., Quantifier-free versions of first-order logic and their psychological significance, Journal of Philosophical Logic, vol. 21 (1992), pp. 125–147.CrossRefGoogle Scholar
[5]Mycielski, J., On the axiom of determinateness. I, II, Fundamenta Mathematicae, vol. 53 (1964), pp. 205–224; Fundamenta Mathematicae, vol. 59 (1966), pp. 203–212.CrossRefGoogle Scholar
[6]Mycielski, J., Locally finite theories, this Journal, vol. 51 (1986), pp. 59–62.Google Scholar
[7]Mycielski, J., Games with perfect information, Chapter 3 in the, Handbook of game theory with economic applications. Vol. 1, North-Holland, Amsterdam, 1992, pp. 41–70.CrossRefGoogle Scholar
[8]Mycielski, J. and Steinhaus, H., A mathematical axiom contradicting the axiom of choice, Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 10 (1962), pp. 1–3.Google Scholar
[9]Myhill, J. and Scott, D. S., Ordinal definability, Axiomatic set theory, Proceedings of Symposia in Pure Mathematics, vol. XIII, Part 1, American Mathematical Society, Providence, Rhode Island, 1971, pp. 271–278.CrossRefGoogle Scholar
[10]Paris, J. B., Minimal models of ZF, The Proceedings of the Bertrand Russell Memorial Conference (Uldum Denmark, 1971), Bertrand Russell Memorial Logic Conference, Leeds, 1973, pp. 327–331.Google Scholar