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GAMES AND RAMSEY-LIKE CARDINALS

Published online by Cambridge University Press:  30 January 2019

DAN SAATTRUP NIELSEN  and
PHILIP WELCH
DAN SAATTRUP NIELSEN
Affiliation:
SCHOOL OF MATHEMATICSUNIVERSITY OF BRISTOL TYNDALL AVE BRISTOL BS8 1TH, UKE-mail: [email protected]
PHILIP WELCH
Affiliation:
SCHOOL OF MATHEMATICSUNIVERSITY OF BRISTOL TYNDALL AVE BRISTOL BS8 1TH, UKE-mail: [email protected]
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Abstract

We generalise the α-Ramsey cardinals introduced in Holy and Schlicht (2018) for cardinals α to arbitrary ordinals α, and answer several questions posed in that paper. In particular, we show that α-Ramseys are downwards absolute to the core model K for all α of uncountable cofinality, that strategic ω-Ramsey cardinals are equiconsistent with remarkable cardinals and that strategic α-Ramsey cardinals are equiconsistent with measurable cardinals for all α > ω. We also show that the n-Ramseys satisfy indescribability properties and use them to provide a game-theoretic characterisation of completely ineffable cardinals, as well as establishing further connections between the α-Ramsey cardinals and the Ramsey-like cardinals introduced in Gitman (2011), Feng (1990), and Sharpe and Welch (2011).

Keywords

03E3503E4503E55Ramsey-like cardinalslarge cardinalsgamesweakly compact cardinalsineffable cardinalscompletely ineffable cardinalsremarkable cardinalsvirtually measurable cardinalsmeasurable cardinalscore model
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 1 , March 2019 , pp. 408 - 437
Copyright
Copyright © The Association for Symbolic Logic 2019 

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References

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