Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-25T11:16:42.423Z Has data issue: false hasContentIssue false
  • English
  • Français

Unification of universes in set theory

Published online by Cambridge University Press:  12 March 2014

W. V. Quine
W. V. Quine*
Affiliation:
Harvard University
Get access Rights & Permissions [Opens in a new window]

Extract

The logic of quantification is conveniently developed in terms of the so-called quantificational schemata. The atomic quantificational schemata consist each of a schematic predicate letter (‘F’, ‘G’, etc.) attached to one or more quantifiable variables (‘x’, ‘y’, etc.). Compound quantificational schemata are built up of these atomic ones by means of truth functions and quantifiers. The business of the logic of quantification then becomes that of spotting the valid schemata: the ones that are true under all interpretations.

The logic of quantification is well understood and well behaved. Proof procedures in this domain are known, thanks to Gödel, to be complete. If decision procedures are not known for the whole of this part of logic, still we do know, thanks to Church, that they cannot exist. And we know how to streamline our proof procedure for quick success in the average case where proof is possible. Because of all this, there is a premium on adapting special theories, of special subject matters, to this general mold. This means representing the truths of the special theory as a class of quantificational schemata with interpreted predicate letters and a chosen universe of discourse. Once a theory has been thus regimented, it is a standard theory (in approximately Tarski's sense of the term).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 21 , Issue 3 , September 1956 , pp. 267 - 279
Copyright
Copyright © Association for Symbolic Logic 1956

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]Bernays, Paul, A system of axiomatic set theory, this Journal, vol. 2 (1937), pp. 65–77. Continued in vols. 6–8, 13, 19.Google Scholar
[2]Church, Alonzo, A formulation of the simple theory of types, this Journal, vol. 5 (1940), pp. 56–68.Google Scholar
[3]Fraenkel, A., Untersuchungen über die Grundlagen der Mengenlehre, Mathematische Zeitschrift, vol. 22 (1925), pp. 250–273.CrossRefGoogle Scholar
[4]Frege, Gottlob, Grundgesetze der Arithmetic, vol. 1, Jena, 1893.Google Scholar
[5]Hailperin, Theodore, A set of axioms for logic, this Journal, vol. 9 (1944), pp. 1–19.Google Scholar
[6]Quine, W. V., Mathematical logic, revised edition. Cambridge, Mass., 1951.CrossRefGoogle Scholar
[7]Quine, W. V., New foundations for mathematical logic, American mathematical monthly, vol. 44 (1937), pp. 70–80. Reprinted with additions in Quine, From a logical point of view (Cambridge, Mass., 1953).CrossRefGoogle Scholar
[8]Quine, W. V., On Cantor's theorem, this Journal, vol. 2 (1937), pp. 120–124.Google Scholar
[9]Rosser, J. B., Logic for mathematicians, New York, 1953.Google Scholar
[10]Specker, Ernst, The axiom of choice in Quine's new foundations for mathematical logic, Proceedings of the National Academy of Sciences, vol. 39 (1953), pp. 972–975.CrossRefGoogle Scholar
[11]Tarski, Alfred, Sur le terme primitif dans la logistique, Fundamenta mathematicae, vol. 4 (1923), pp. 196–200.Google Scholar
[12]Zermelo, Ernst, Untersuchungen über die Grundlagen der Mengenlehre, Mathematische Annalen, vol. 65 (1908), pp. 261–281.CrossRefGoogle Scholar