Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-16T16:16:11.467Z Has data issue: false hasContentIssue false
  • English
  • Français

Eliminating the continuum hypothesis

Published online by Cambridge University Press:  12 March 2014

Richard A. Platek
Richard A. Platek*
Affiliation:
Cornell University
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we show how the assumption of the generalized continuum hypothesis (GCH) can be removed or partially removed from proofs in Zermelo-Frankel set theory (ZF) of statements expressible in the simple theory of types. We assume the reader is familiar with the latter language, especially with the classification of formulas and sentences of that language into Σκη and Πκη form (cf. [1]) and with how that language can be relatively interpreted into the language of ZF.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 34 , Issue 2 , 25 July 1969 , pp. 219 - 225
Copyright
Copyright © Association for Symbolic Logic 1969

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.)

Footnotes

1

The results described in this paper were obtained while the author was a C. L. E. Moore Instructor at M.I.T. The paper was prepared under National Science Foundation grant GP 8732.

References

[1] Shoenfield, J. R., Mathematical logic, Addison-Wesley, Reading, Massachusetts, 1967.Google Scholar
[2] Shoenfield, J. R., The problem of predicativity, Essays on the foundations of mathematics, Magnes Press, Jerusalem, 1961, pp. 132–139.Google Scholar
[3] Platek, R. A., Eliminating the continuum hypothesis, Notices of the American Mathematical Society, vol. 14 (1967), p. 366.Google Scholar
[4] Feferman, S., Independence results in set theory by Cohen's method. II, Notices of the American Mathematical Society, vol. 10 (1963), p. 593.Google Scholar
[5] Shoenfield, J. R., On the independence of the axiom of constructibility, American Journal of Mathematics, vol. 81 (1959), pp. 537–540.Google Scholar
[6] Hajnal, A., On a consistency theorem connected with the generalized continuum hypothesis, Acta. Mathematica Academiae Scientiarum Hungaricae, vol. 12 (1961), pp. 321–376.Google Scholar
[7] Cohen, P. J., The independence of the continuum hypothesis, Proceedings of the National Academy of Sciences, vol. 50 (1963), pp. 1143–1148 and vol. 51 (1964), pp. 105-110.Google Scholar
[8] Solovay, R., 2ℵo can be anything it ought to be, The theory of models, North-Holland, Amsterdam, 1965, p. 435.Google Scholar