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Archive for Mathematical Logic

, Volume 56, Issue 3–4, pp 403–421 | Cite as

A strong partition cardinal above \(\varTheta \)

  • Daniel W. CunninghamEmail author
Article
  • 55 Downloads

Abstract

Assuming \(\text {ZF}+\text {DC}\), we prove that if there exists a strong partition cardinal greater than \(\varTheta \), then (1) there is an inner model of \(\text {ZF}+\text {AD}+\text {DC}+ {{{\mathbb {R}}} }^{{\#}}\) exists, and (2) there is an inner model of \(\text {ZF}+\text {AD}+\text {DC}+ (\exists \kappa >\varTheta )\,(\kappa \) is measurable). Here \(\varTheta \) is the supremum of the ordinals which are the surjective image of the set of reals \({{{\mathbb {R}}} }\).

Keywords

Strong partition cardinals Determinacy Inner models 

Mathematics Subject Classification

03E15 03E60 03E02 03E45 

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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Mathematics DepartmentSUNY Buffalo StateBuffaloUSA

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