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Large cardinals and definable well-orders on the universe

Published online by Cambridge University Press:  12 March 2014

Andrew D. Brooke-Taylor
Andrew D. Brooke-Taylor*
Affiliation:
Department of Mathematics, University of Bristol, University Walk, Bristol, Bs8 1Tw, UK, E-mail: Andrew.Brooke-Taylor@bristol.ac.uk
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Abstract

We use a reverse Easton forcing iteration to obtain a universe with a definable well-order, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle , at a proper class of cardinals κ. By choosing the cardinals at which coding occurs sufficiently sparsely, we are able to lift the embeddings witnessing the large cardinal properties without having to meet any non-trivial master conditions.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 2 , June 2009 , pp. 641 - 654
Copyright
Copyright © Association for Symbolic Logic 2009

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References

REFERENCES

[1]Asperó, David and Friedman, Sy-David, Large cardinals and locally defined well-orders of the universe, Annals of Pure and Applied Logic, vol. 157 (2009). no. 1, pp. 1–15.CrossRefGoogle Scholar
[2]Brooke-Taylor, Andrew D., Large cardinals and L-like combinatorics, Ph.D. thesis, University of Vienna, 06 2007.Google Scholar
[3]Brooke-Taylor, Andrew D. and Friedman, Sy-David, Large cardinalsandgap-1 morasses, Annals of Pure and Applied Logic, to appear.Google Scholar
[4]Burke, Douglas, Generic embeddings and the failure of box, Proceedings of the American Mathematical Society, vol. 123 (1995), no. 9. pp. 2867–2871.CrossRefGoogle Scholar
[5]Cummings, James, Iterated forcing and elementary embeddings, The handbook of set theory (Foreman, Matthew and Kanamori, Akihiro, editors), vol. II, Springer, 2009, to appear.Google Scholar
[6]Cummings, James, Foreman, Matthew, and Magidor, Menachem, Squares, scales and stationary reflection, Journal of Mathematical Logic, vol. 1 (2001), no. 1, pp. 35–98.CrossRefGoogle Scholar
[7]Cummings, James and Schimmerling, Ernest, Indexed squares, Israel Journal of Mathematics, vol. 131 (2002), pp. 61–99.CrossRefGoogle Scholar
[8]Devlin, Keith J., Variations on ⟡, this Journal, vol. 44 (1979), no. 1, pp. 51–58.Google Scholar
[9]Friedman, Sy D., Fine structure and class forcing, de Gruyter Series in Logic and Its Applications, no. 3, de Gruyter, Berlin, 2000.CrossRefGoogle Scholar
[10]Friedman, Sy D., Large cardinals and L-like universes, Set theory: Recent trends and applications (Andretta, Alessandro, editor), Quadernidi Matematica. vol. 17. Seconda Università di Napoli, 2005, pp. 93–110.Google Scholar
[11]Friedman, Sy D., Forcing condensation, preprint.Google Scholar
[12]Hamkins, Joel David, Fragile measurability, this Journal, vol. 59 (1994), no. 1, pp. 262–282.Google Scholar
[13]Hamkins, Joel David, Gup forcing: generalizing the Levy–Solovay theorem, The Bulletin of Symbolic Logic, vol. 5 (1999). no. 2, pp. 264–272.CrossRefGoogle Scholar
[14]Hamkins, Joel David, The lottery prepartaion, Annals of Pure and Applied Logic, vol. 101 (2000), no. 2–3, pp. 103–146.CrossRefGoogle Scholar
[15]Hamkins, Joel David, The wholeness axioms and V = HOD, Archive for Mathematical Logic, vol. 40 (2001). no. 1, pp. 1–8.CrossRefGoogle Scholar
[16]Jech, Thomas, Set theory, third millenium ed., Springer, 2003.Google Scholar
[17]Jensen, Ronald Björn, Measurable cardinals and the GCH, Proceedings of Symposia in Pure Mathematics (Jech, Thomas J.. editor). Axiomatic set theory, vol. 13. part II. American Mathematical Society, 1974, pp. 175–178.Google Scholar
[18]Kanamori, Akihiro, The higher infinite, second ed., Springer, 2003.Google Scholar
[19]Kunen, Kenneth, Elementary embeddings and infinitary combinatorics, this Journal, vol. 36 (1971), pp. 407–413.Google Scholar
[20]Kunen, Kenneth, Set theory, North-Holland, 1980.Google Scholar
[21]Law, David. An abstract condensation property, Ph.D. thesis. California Institute of Technology, 1993.Google Scholar
[22]McAloon, Kenneth. Consistency results about ordinal definability, Annals of Mathematical Logic, vol. 2 (1970/1971). no. 4. pp. 449–467.CrossRefGoogle Scholar
[23]Menas, Telis K., Consistency results concerning supercompactness. Transactions of the American Mathematical Society, vol. 223 (1976), pp. 61–91.CrossRefGoogle Scholar
[24]Velleman, Daniel J., Morasses, diamond, and forcing, Annals of Mathematical Logic, vol. 23 (1982), no. 23. pp. 199–281.Google Scholar
[25]Woodin, W. Hugh, The axiom of determinacy, forcing axioms, and the nonstationary ideal, de Gruyter, 1999.CrossRefGoogle Scholar