Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-22T20:11:48.606Z Has data issue: false hasContentIssue false
  • English
  • Français

A formal system of logic

Published online by Cambridge University Press:  12 March 2014

Hao Wang
Hao Wang*
Affiliation:
Harvard University
Get access Rights & Permissions [Opens in a new window]

Extract

The main purpose of this paper is to present a formal system P in which we enjoy a smooth-running technique and which countenances a universe of classes which is symmetrical as between large and small. More exactly, P is a system which differs from the inconsistent system of [1] only in the introduction of a rather natural new restrictive condition on the defining formulas of the elements (sets, membership-eligible classes). It will be proved that if the weaker system of [2] is consistent, then P is also consistent.

After the discovery of paradoxes, it may be recalled, Russell and Zermelo in the same year proposed two different ways of safeguarding logic against contradictions (see [3], [4]). Since then various simplifications and refinements of these systems have been made. However, in the resulting systems of Zermelo set theory, generation of classes still tends to be laborious and uncertain; and in the systems of Russell's theory of types, complications in the matter of reduplication of classes and meaningfulness of formulas remain. In [2], Quine introduced a system which seems to be free from all these complications. But later it was found out that in it there appears to be an unavoidable difficulty connected with mathematical induction. Indeed, we encounter the curious situation that although we can prove in it the existence of a class V of all classes, and we can also prove particular existence theorems for each of infinitely many classes, nobody has so far contrived to prove in it that V is an infinite class or that there exists an infinite class at all.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 15 , Issue 1 , March 1950 , pp. 25 - 32
Copyright
Copyright © Association for Symbolic Logic 1951

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]Quine, W. V.. Mathematical logic. New York, 1940.Google Scholar
[2]Quine, W. V.. New foundations for mathematical logic, American mathematical monthly, vol. 44 (1937), pp. 70–80.CrossRefGoogle Scholar
[3]Russell, Bertrand. Mathematical logic based on the theory of types, American journal of mathematics, vol. 30 (1908), pp. 222–262.CrossRefGoogle Scholar
[4]Zermelo, Ernst. Untersuchungen über die Grundlagen der Mengenlehre I, Mathematische Annalen, vol. 65 (1908), pp. 261–281.CrossRefGoogle Scholar
[5]Rosser, Barkley. On the consistency of Quine's “New foundations for mathematical logic,” this Journal, vol. 4 (1939), pp. 15–24.Google Scholar
[6]Rosser, Barkley. Definition by induction in Quine's “New foundations for mathematical logic,” Mathematische Annalen, pp. 80–81.Google Scholar
[7]von Neumann, J.. Eine Axiomatisierung der Mengenlehre, Journal für reine und angewandte Mathematik, vol. 154 (1925), pp. 219–240.CrossRefGoogle Scholar
[8]Bernays, Paul. A system of axiomatic set theory, Part I, this Journal, vol. 2 (1937), pp. 65–77; Part II, ibid., vol. 6 (1941), pp. 1-17.Google Scholar
[9]Gödel, Kurt. The consistency of the continuum hypothesis. Princeton, 1940.CrossRefGoogle Scholar
[10]Rosser, Barkley. The Burali-Forti paradox, this Journal, vol. 7 (1942), pp. 1–17.Google Scholar
[11]Quine, W. V.. Element and number, this journal, vol. 6 (1941), pp. 135–149.Google Scholar
[12]Wang, Hao. A new theory of element and number, this journal, vol. 13 (1948), pp. 129–137.Google Scholar
[13]Fitch, F. B.. Review of [11], this journal, vol. 7 (1942), pp. 121–122.Google Scholar
[14]Black, Max. Review of [1], Mind, n. s., vol. 52 (1943), pp. 264–275.CrossRefGoogle Scholar
[15]Quine, W. V.. On existence conditions for elements and classes, this Journal, vol. 7 (1942), pp. 157–159.Google Scholar
[16]Wang, Hao. Existence of classes and value specification of variables, this Journal. (To appear soon.)Google Scholar
[17]Whitehead, A. N. and Russell, Bertrand. Principia mathematica. Cambridge, England, 1910–1913.Google Scholar
[18]Gödel, Kurt. Die Vollständigkeit der Axiome des logischen Funktionenkalküls, Monatshefte für Mathematik und Physik, vol. 37 (1930), pp. 349–360.CrossRefGoogle Scholar
[19]Church, Alonzo. Introduction to mathematical logic. Princeton, 1944.Google Scholar
[20]Wang, Hao. On Zermelo's and von Neumann's axioms for set theory, Proceedings of the National Academy of Sciences of the United States of America, vol. 35 (1949), pp. 150–155.CrossRefGoogle ScholarPubMed