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Large cardinals and definable well-orders on the universe

  • Andrew D. Brooke-Taylor (a1)


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.



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[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. 115.
[2]Brooke-Taylor, Andrew D., Large cardinals and L-like combinatorics, Ph.D. thesis, University of Vienna, 06 2007.
[3]Brooke-Taylor, Andrew D. and Friedman, Sy-David, Large cardinalsandgap-1 morasses, Annals of Pure and Applied Logic, to appear.
[4]Burke, Douglas, Generic embeddings and the failure of box, Proceedings of the American Mathematical Society, vol. 123 (1995), no. 9. pp. 28672871.
[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.
[6]Cummings, James, Foreman, Matthew, and Magidor, Menachem, Squares, scales and stationary reflection, Journal of Mathematical Logic, vol. 1 (2001), no. 1, pp. 3598.
[7]Cummings, James and Schimmerling, Ernest, Indexed squares, Israel Journal of Mathematics, vol. 131 (2002), pp. 6199.
[8]Devlin, Keith J., Variations on ⟡, this Journal, vol. 44 (1979), no. 1, pp. 5158.
[9]Friedman, Sy D., Fine structure and class forcing, de Gruyter Series in Logic and Its Applications, no. 3, de Gruyter, Berlin, 2000.
[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. 93110.
[11]Friedman, Sy D., Forcing condensation, preprint.
[12]Hamkins, Joel David, Fragile measurability, this Journal, vol. 59 (1994), no. 1, pp. 262282.
[13]Hamkins, Joel David, Gup forcing: generalizing the Levy–Solovay theorem, The Bulletin of Symbolic Logic, vol. 5 (1999). no. 2, pp. 264272.
[14]Hamkins, Joel David, The lottery prepartaion, Annals of Pure and Applied Logic, vol. 101 (2000), no. 2–3, pp. 103146.
[15]Hamkins, Joel David, The wholeness axioms and V = HOD, Archive for Mathematical Logic, vol. 40 (2001). no. 1, pp. 18.
[16]Jech, Thomas, Set theory, third millenium ed., Springer, 2003.
[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. 175178.
[18]Kanamori, Akihiro, The higher infinite, second ed., Springer, 2003.
[19]Kunen, Kenneth, Elementary embeddings and infinitary combinatorics, this Journal, vol. 36 (1971), pp. 407413.
[20]Kunen, Kenneth, Set theory, North-Holland, 1980.
[21]Law, David. An abstract condensation property, Ph.D. thesis. California Institute of Technology, 1993.
[22]McAloon, Kenneth. Consistency results about ordinal definability, Annals of Mathematical Logic, vol. 2 (1970/1971). no. 4. pp. 449467.
[23]Menas, Telis K., Consistency results concerning supercompactness. Transactions of the American Mathematical Society, vol. 223 (1976), pp. 6191.
[24]Velleman, Daniel J., Morasses, diamond, and forcing, Annals of Mathematical Logic, vol. 23 (1982), no. 23. pp. 199281.
[25]Woodin, W. Hugh, The axiom of determinacy, forcing axioms, and the nonstationary ideal, de Gruyter, 1999.

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Large cardinals and definable well-orders on the universe

  • Andrew D. Brooke-Taylor (a1)


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