Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T19:02:01.238Z Has data issue: false hasContentIssue false
  • English
  • Français

A ${\rm{\Sigma }}_4^1 $ WELLORDER OF THE REALS WITH ${\rm{NS}}_{\omega _1 } $ SATURATED

Published online by Cambridge University Press:  16 July 2019

SY-DAVID FRIEDMAN  and
STEFAN HOFFELNER
SY-DAVID FRIEDMAN
Affiliation:
KURT GÖDEL RESEARCH CENTER UNIVERSITÄT WIEN WIEN, AUSTRIA E-mail: sdf@logic.univie.ac.at
STEFAN HOFFELNER
Affiliation:
WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTER MÜNSTER, GERMANY E-mail: hoffelne@uni-muenster.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that, assuming the existence of the canonical inner model with one Woodin cardinal $M_1 $ , there is a model of $ZFC$ in which the nonstationary ideal on $\omega _1 $ is $\aleph _2 $-saturated and whose reals admit a ${\rm{\Sigma }}_4^1 $-wellorder.

Keywords

03E3503E4503E4703E55nonstationary idealprojective wellorderssaturation
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 4 , December 2019 , pp. 1466 - 1483
Copyright
Copyright © The Association for Symbolic Logic 2019 

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

Caicedo, A. and Velickovic, B., The bounded proper forcing axiom and wellorderings of the reals. Mathematical Research Letters, vol. 13 (2006), no. 2–3, pp. 393–408.CrossRefGoogle Scholar
Friedman, S. D. and Wu, L., Large cardinals and the ${\rm{\Delta }}_1 $-definability of the nonstationary ideal, to appear.Google Scholar
Gitik, M. and Shelah, S., Less saturated ideals. Proceedings of the American Mathematical Society, vol. 125 (1997), no. 5, pp. 1523–1530.CrossRefGoogle Scholar
Hjorth, G., The size of the ordinal $u_2 $ . Journal of the London Mathematical Society (2), vol. 52 (1995), no. 3, pp. 417–433.CrossRefGoogle Scholar
Jensen, R. and Steel, J., K without a measurable, this Journal, vol. 78 (2013), no. 3, pp. 708–734.Google Scholar
Kunen, K., Saturated ideals, this Journal, vol. 43 (1978), no. 1, pp. 65–76.Google Scholar
Schlindwein, C., Simplified RCS iterations. Archive for Mathematical Logic, vol. 32 (1993), no. 5, pp. 341–349.CrossRefGoogle Scholar
Schindler, R., On $NS_{\omega _1 } $ being saturated. Online Notes , 2016. Available at http://www.math.uni-muenster.de/u/rds/sat_ideal_better_version.pdf.Google Scholar
Steel, J., Inner models with many Woodin cardinals. Annals of Pure and applied Logic, vol. 65 (1993), pp. 185–209.CrossRefGoogle Scholar
Steel, J., Projectively wellordered inner models. Annals of Pure and Applied Logic, vol. 74 (1995), no. 1, pp. 77–104.CrossRefGoogle Scholar
Woodin, H. W., The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, De Gruyter, Berlin 2001.Google Scholar