Skip to main content Accessibility help
×
Home

Incompatible Ω-Complete Theories

  • Peter Koellner (a1) and W. Hugh Woodin (a2)

Abstract

In 1985 the second author showed that if there is a proper class of measurable Woodin cardinals and and are generic extensions of V satisfying CH then and agree on all Σ12-statements. In terms of the strong logic Ω-logic this can be reformulated by saying that under the above large cardinal assumption ZFC + CH is Ω-complete for Σ12. Moreover, CH is the unique Σ12-statement with this feature in the sense that any other Σ12-statement with this feature is Ω-equivalent to CH over ZFC. It is natural to look for other strengthenings of ZFC that have an even greater degree of Ω-completeness. For example, one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for all of third-order arithmetic. Going further, for each specifiable segment Vλ of the universe of sets (for example, one might take Vλ to be the least level that satisfies there is a proper class of huge cardinals), one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for the theory of Vλ. If such theories exist, extend one another, and are unique in the sense that any other such theory B with the same level of Ω-completeness as A is actually Ω-equivalent to A over ZFC, then this would show that there is a unique Ω-complete picture of the successive fragments of the universe of sets and it would make for a very strong case for axioms complementing large cardinal axioms. In this paper we show that uniqueness must fail. In particular, we show that if there is one such theory that Ω-implies CH then there is another that Ω-implies ¬-CH.

Copyright

References

Hide All
[1]Abraham, Uri and Shelah, Saharon, Δ22 well-order of the reals and incompactness of L(QMM), Annals of Pure and Applied Logic, vol. 59 (1993), no. 1, pp. 132.
[2]Bagaria, Joan, Castells, Neus, and Larson, Paul, An Ω-logic primer, Set theory (Bagaria, Joan and Todorcevic, Stevo, editors), Trends in Mathematics, Birkhäuser, Basel, 2006, pp. 128.
[3]Davis, Morton, Infinite games of perfect information, Advances in game theory (Dresher, Melvin, Shapley, Lloyd S, and Tucker, Alan W., editors), Annals of Mathematical Studies, vol. 52, Princeton University Press, Princeton, 1964, pp. 85101.
[4]Feferman, Solomon Jr., Dawson, John W., Kleene, Stephen C., Moore, Gregory H., Solovay, Robert M., and van Heijenoort, Jean (editors), Gödel, Kurt, Collected works, Volume II: Publications 1938–1974, Oxford University Press, New York and Oxford, 1990.
[5]Feng, Qi, Magidor, Menacham, and Woodin, W. Hugh, Universally Baire sets of reals, Set theory of the continuum (Judah, Haim, Just, Winfried, and Woodin, W. Hugh, editors), Mathematical Sciences Research Institute, vol. 26, Springer-Verlag, Berlin, 1992, pp. 203242.
[6]Gödel, Kurt, Remarks before the Princeton bicentennial conference on problems in mathematics, In Feferman et al. [4], pp. 150153.
[7]Hamkins, Joel David and Woodin, W. Hugh, Small forcing creates neither strong nor Woodin cardinals, Proceedings of the American Mathematical Society, vol. 128 (2000), no. 10, pp. 30253029.
[8]Kanamori, Akihiro, The higher infinite: Large cardinals in set theory from their beginnings, second ed., Springer Monographs in Mathematics, Springer, Berlin, 2003.
[9]Koellner, Peter, On the question of absolute undecidability, Philosophia Mathematica, vol. 14 (2006), no. 2, pp. 153188, Revised and reprinted in Kurt Gödel: Essays for his Centennial, edited by Solomon Feferman, Charles Parsons, and Stephen G. Simpson. Lecture Notes in Logic, 33. Association of Symbolic Logic, 2009.
[10]Koellner, Peter, Truth in mathematics: The question of pluralism. New waves in philosophy of mathematics (Bueno, Otávio and Linnebo, Øystein, editors), New Waves in Philosophy, Palgrave Macmillan, 2009, Forthcoming.
[11]Larson, Paul, The stationary tower: Notes on a course by W. Hugh Woodin, University Lecture Series, vol. 32, American Mathematical Society, 2004.
[12]Larson, Paul, Ketchersid, Richard, and Zapletal, Jindrich, Regular embeddings of the stationary tower and Woodin's Σ22 maximality theorem, preprint, 2008.
[13]Laver, Richard, Certain very large cardinals are not created in small forcing extensions. Annals of Pure and Applied Logic, vol. 149 (2007), no. 1-3, pp. 16.
[14]Lévy, Azriel and Solovay, Robert M., Measurable cardinals and the continuum hypothesis, Israel Journal of Mathematics, vol. 5 (1967), pp. 234248.
[15]Martin, Donald A. and Steel, John R., The extent of scales in L(ℝ), Cabal seminar 79–81 (Kechris, Alexander S., Martin, Donald A., and Moschovakis, Yiannis S., editors), Lecture Notes in Mathematics, no. 1019, Springer-Verlag, Berlin, 1983, pp. 8696.
[16]Martin, Donald A. and Steel, John R., A proof of projective determinacy, Journal of the American Mathematical Society, vol. 2 (1989), no. 1, pp. 71125.
[17]Mycielski, Jan and Swierczkowski, Stanislaw, On the Lebesgue measurability and the axiom of determinateness, Fundament a Mathematicae, vol. 54 (1964), pp. 6771.
[18]Paris, Jeff and Harrington, Leo, A mathematical incompleteness in Peano Arithmetic, Handbook of mathematical logic (Barwise, Jon, editor), Studies in Logic and the Foundations of Mathematics, vol. 90, Elsevier, Amsterdam, 1977, pp. 11331142.
[19]Shelah, Saharon and Woodin, W. Hugh, Large cardinals imply that every reasonably definable set of reals is Lebesgue measurable, Israel Journal of Mathematics, vol. 70 (1990), pp. 381394.
[20]Woodin, W. Hugh, Σ12 absoluteness, unpublished, 05 1985.
[21]Woodin, W. Hugh,Supercompact cardinals, sets of reals, and weakly homogeneous trees, Proceedings of the National Academy of Sciences, vol. 85 (1988), no. 18, pp. 65876591.
[22]Woodin, W. Hugh,The axiom of determinacy, forcing axioms, and the nonstationary ideal, de Gruyter, Series in Logic and its Applications, vol. 1, de Gruyter, Berlin, 1999.
[23]Woodin, W. Hugh,Beyond absoluteness, Proceedings of the International Congress of Mathematicians, (Beijing, 2002), vol. I, Higher Education Press, Beijing, 2002, pp. 515524.
[24]Woodin, W. Hugh,Suitable Extender Sequences, To appear, 2009.

Related content

Powered by UNSILO

Incompatible Ω-Complete Theories

  • Peter Koellner (a1) and W. Hugh Woodin (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.