Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-20T02:34:07.318Z Has data issue: false hasContentIssue false
  • English
  • Français

LARGE CARDINALS AND LIGHTFACE DEFINABLE WELL-ORDERS, WITHOUT THE GCH

Published online by Cambridge University Press:  13 March 2015

SY-DAVID FRIEDMAN ,
PETER HOLY  and
PHILIPP LÜCKE
SY-DAVID FRIEDMAN
Affiliation:
UNIVERSITÄT WIEN, KURT GÖDEL RESEARCH CENTER FOR MATHEMATICAL LOGIC, WÄHRINGER STRASSE 25, 1090 WIEN, AUSTRIAE-mail: [email protected]
PETER HOLY
Affiliation:
UNIVERSITY OF BRISTOL, DEPARTMENT OF MATHEMATICS, UNIVERSITY WALK, BRISTOL BS8 1TW, UKE-mail: [email protected]
PHILIPP LÜCKE
Affiliation:
RHEINISCHE FRIEDRICH-WILHELMS-UNIVERSITÄT BONN, MATHEMATISCHES INSTITUT, ENDENICHER ALLEE 60, 53115 BONN, GERMANYE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper deals with the question whether the assumption that for every inaccessible cardinal κ there is a well-order of H(κ+) definable over the structure $\langle {\rm{H}}({\kappa ^ + }), \in \rangle$ by a formula without parameters is consistent with the existence of (large) large cardinals and failures of the GCH. We work under the assumption that the SCH holds at every singular fixed point of the ℶ-function and construct a class forcing that adds such a well-order at every inaccessible cardinal and preserves ZFC, all cofinalities, the continuum function, and all supercompact cardinals. Even in the absence of a proper class of inaccessible cardinals, this forcing produces a model of “V = HOD” and can therefore be used to force this axiom while preserving large cardinals and failures of the GCH. As another application, we show that we can start with a model containing an ω-superstrong cardinal κ and use this forcing to build a model in which κ is still ω-superstrong, the GCH fails at κ and there is a well-order of H(κ+) that is definable over H(κ+) without parameters. Finally, we can apply the forcing to answer a question about the definable failure of the GCH at a measurable cardinal.

Keywords

03E3503E4703E55Class forcingDefinable well-ordersFailures of the GCHHODsupercompact cardinalsω-superstrong cardinals
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 1 , March 2015 , pp. 251 - 284
Copyright
Copyright © The Association for Symbolic Logic 2015 

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

Apter, Arthur W. and Friedman, Shoshana, Coding into HOD via normal measures with some applications. Mathematical Logic Quarterly, vol. 57 (2011), no. 4, pp. 366–372.Google Scholar
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
Asperó, David and Friedman, Sy-David, Definable well-orders of H(ω 2) and GCH. this Journal, vol. 77 (2012), no. 4, pp. 1101–112.Google Scholar
Asperó, David, Holy, Peter, and Lücke, Philipp, Forcing lightface definable well-orders without GCH, submitted.Google Scholar
Brooke-Taylor, Andrew D., Large cardinals and definable well-orders on the universe. this Journal, vol. 74 (2009), pp. 641–654.Google Scholar
Friedman, Shoshana, Accessing the switchboard via set forcing. Mathematical Logic Quarterly, vol. 58 (2012), no. (4-5), pp. 303–306.CrossRefGoogle Scholar
Friedman, Sy-David, Fine structure and class forcing. de Gruyter Series in Logic and its Applications, vol. 3, Walter de Gruyter & Co., Berlin, 2000.CrossRefGoogle Scholar
Friedman, Sy-David, Large cardinals and L-like universes. Set theory: recent trends and applications, vol. 17, Quad. Mat, Dept. Math., Seconda Univ. Napoli, Caserta, 2006, pp. 93–110.Google Scholar
Friedman, Sy-David and Holy, Peter, Condensation and large cardinals. Fundamenta Mathematicae, vol. 215 (2011), no. 2, pp. 133–166.Google Scholar
Friedman, Sy-David and Honzik, Radek, A definable failure of the singular cardinal hypothesis. Israel Journal of Mathematics, Vol. 192 (2012), no. 2, pp 719–762.CrossRefGoogle Scholar
Friedman, Sy-David and Lücke, Philipp, Large cardinals and definable well-orders, without the GCH. Annals of Pure and Applied Logic, accepted.Google Scholar
Kanamori, A. and Magidor, M., The evolution of large cardinal axioms in set theory. Higher set theory, Proc. Conference Math. Forschungsinst., Oberwolfach, 1977, vol. 669, Lecture Notes in Mathematics, Springer, Berlin, 1978, pp. 99–275.Google Scholar
Lücke, Philipp, $\Sigma _1^1$-definability at uncountable regular cardinals. this Journal, vol. 77 (2012), no. 3, pp. 1011–1046.Google Scholar