English

Games and Ramsey-like cardinals

Logic 2018-10-31 v3

Abstract

We generalise the α\alpha-Ramsey cardinals introduced in Holy and Schlicht (2018) for cardinals α\alpha to arbitrary ordinals α\alpha, and answer several questions posed in that paper. In particular, we show that α\alpha-Ramseys are downwards absolute to the core model KK for all α\alpha of uncountable cofinality, that strategic ω\omega-Ramsey cardinals are equiconsistent with remarkable cardinals and that strategic α\alpha-Ramsey cardinals are equiconsistent with measurable cardinals for all α>ω\alpha>\omega. We also show that the nn-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\alpha-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