Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-11T08:05:46.430Z Has data issue: false hasContentIssue false
  • English
  • Français

Canonical seeds and Prikry trees

Published online by Cambridge University Press:  12 March 2014

Joel David Hamkins
Joel David Hamkins*
Affiliation:
Mathematics 15-215, City University of New York, CSI, 2800 Victory Blvd., Staten Island, NY 10314, USA, E-mail: hamkins@integral.math.csi.cuny.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Applying the seed concept to Prikry tree forcing ℙμ, I investigate how well ℙμ preserves the maximality property of ordinary Prikry forcing and prove that ℙμ, Prikry sequences are maximal exactly when μ admits no non-canonical seeds via a finite iteration. In particular, I conclude that if μ is a strongly normal supercompactness measure, then ℙμ Prikry sequences are maximal, thereby proving, for a large class of measures, a conjecture of W. Hugh Woodin's.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 2 , June 1997 , pp. 373 - 396
Copyright
Copyright © Association for Symbolic Logic 1997

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]Blass, Andreas, Selective ultrafilters and homogeneity, Annals of Pure and Applied Logic, vol. 38 (1988), pp. 215–255.CrossRefGoogle Scholar
[2]Cummings, James and Woodin, Hugh, Generalized Prikry forcings, preprint, 1990.Google Scholar
[3]Dehornoy, Patrick, Iterated ultrapowers and Prikry forcing, Annals of Mathematical Logic, vol. 15 (1978), pp. 109–160.CrossRefGoogle Scholar
[4]Gitik, , All uncountable cardinals can be singular, Israel Journal of Mathematics, vol. 35, pp. 61–88.CrossRefGoogle Scholar
[5]Hamkins, Joel David, Lifting and extending measures; fragile measurability, Ph.D. thesis, UC Berkeley, 1994.CrossRefGoogle Scholar
[6]Henle, J. M., Partition properties and Prikry forcing on simple spaces, this Journal, vol. 55 (1990), pp. 938–947.Google Scholar
[7]Jech, Thomas, Set theory, Academic Press, New York, 1978.Google Scholar
[8]Kafkoulis, George, The consistency strength of an infinitary Ramsey property, this Journal, vol. 59 (1994), pp. 1158–1195.Google Scholar
[9]Kanamori, Akihiro, Large cardinals in set theory, Springer-Verlag, Berlin, 1994.Google Scholar
[10]Kunen, Kenneth, Some applications of iterated ultrapowers in set theory, Annals of Mathematical Logic (1970), pp. 179–227.Google Scholar
[11]Louveau, A., Une méthode topologique pour l'étude de la propriété de Ramsey, Israel Journal of Mathematics, vol. 23 (1976), pp. 97–116.CrossRefGoogle Scholar
[12]Mathias, Adrian, On sequences generic in the sense of Prikry, Journal of the Australian Mathematical Society, vol. 15 (1973), pp. 409–414.CrossRefGoogle Scholar
[13]Menas, Telis K., A combinatorial property of Pkλ, this Journal, vol. 41 (1976), pp. 225–234.Google Scholar
[14]Prikry, Karel L., Changing measurable into accessible cardinals, Dissertationes Mathematicae (Roszprawy Matematyczne), vol. 68 (1970), pp. 5–52.Google Scholar