Games and Ramsey-like cardinals
Abstract
We generalise the -Ramsey cardinals introduced in Holy and Schlicht (2018) for cardinals to arbitrary ordinals , and answer several questions posed in that paper. In particular, we show that -Ramseys are downwards absolute to the core model for all of uncountable cofinality, that strategic -Ramsey cardinals are equiconsistent with remarkable cardinals and that strategic -Ramsey cardinals are equiconsistent with measurable cardinals for all . We also show that the -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 -Ramsey cardinals and the Ramsey-like cardinals introduced in Gitman (2011), Feng (1990) and Sharpe and Welch (2011).
Keywords
Cite
@article{arxiv.1804.10383,
title = {Games and Ramsey-like cardinals},
author = {Dan Saattrup Nielsen and Philip Welch},
journal= {arXiv preprint arXiv:1804.10383},
year = {2018}
}
Comments
33 pages, 2 figures. Added Theorem 4.20 saying that strategic $(\omega{+}1)$-Ramsey cardinals are equiconsistent with measurables, and fixed many typos. This version is forthcoming in the JSL