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Easton's Results Via Iterated Boolean-Valued Extensions

Published online by Cambridge University Press:  20 November 2018

Donald H. Pelletier
Donald H. Pelletier*
Affiliation:
York University, Downsview, Ontario
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The purpose of this article is to show how the main result of Easton [1] can be obtained as a special case of a general theory, which was developed in [6], of Boolean-valued models of ZF when the Boolean algebra is a proper class in the ground model. Indeed [1] was the motivating example for [6], Thus the present article together with [6] contain a presentation of Easton's forcing argument in the context of Boolean-valued models. This presentation is not, however, an automatic translation of Easton's argument from the language of forcing to that of Boolean-valued models. In fact, we hope to illuminate the "black magic" referred to in Rosser [8, p. 169].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 4 , August 1974 , pp. 820 - 828
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Easton, W. B., Powers of regular cardinals, Ann. Math. Logic 1 (1970), 139–178.Google Scholar
2. Halmos, P. R., Lectures on Boolean algebras (Van Nostrand, Princeton, 1963).Google Scholar
3. Kunen, K., Inaccessibility properties of cardinals, Ph.D. Thesis, Stanford University, 1968.Google Scholar
4. Levy, A. and Solovay, R. M., Measurable cardinals and the continuum hypothesis, Israel J. Math. 5 (1967), 234–248.Google Scholar
5. McAloon, K., Some applications of Cohen s method, Ph.D. Thesis, University of California, Berkeley, 1966.Google Scholar
6. Pelletier, D. H., Forcing with proper classes: a Boolean-valued approach, Bull. Acad. Polon. Sci Ser. Sci. Math. Astronom. Phys. 22 (1974), 345–352.Google Scholar
7. Pelletier, D. H., Forcing with proper classes, Notices Amer. Math. Soc. 19 (1972), A-19.Google Scholar
8. Rosser, J. B., Simplified independence proofs (Academic Press, New York, 1969).Google Scholar
9. Sikorski, R., Boolean algebras, third edition (Springer-Verlag, New York, 1969).Google Scholar
10. Shoenfield, J. R., Unramified forcing, Proc. Symp. Pure Math., Vol. XIII, Part 1 (1971), 357–381.Google Scholar
11. Scott, D. and Solovay, R., Boolean-valued models of set theory, Proc. Symp. Pure Math., Vol. X I I I , Part 2 (to appear).Google Scholar
12. Solovay, R. and Tennenbaum, S., Iterated Cohen extensions and Souslins problem, Ann. of Math. 94 (1971), 201–245.Google Scholar
13. Takeuti, G., A relativisation of axioms of strong infinity to coi, Ann. Japan Assoc. Philos. Sci. 8 (1970), 191–204.Google Scholar