D. S. Nielsen and P. Welch, Games and Ramsey-like cardinals, 2018, manuscript under review — arXiv.
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 -Ramseys are also strategic -Ramsey, and that strategic -Ramsey cardinals are equiconsistent with measurable cardinals, both by showing that they are measurable in and that they carry precipitous ideals. We also show that the -Ramseys satisfy indescribability properties and use them to characterise ineffable-type cardinals, as well as establishing connections between the -Ramsey cardinals and the Ramsey-like cardinals introduced in Gitman (2011), Feng (1990) and Sharpe and Welch (2011).
The Ramsey-like cardinals can loosely be described as cardinals which are critical points of an elementary embedding , where and are -models of size which contains as both a subset and an element (we call such models weak -models). We can therefore see these cardinals as “mini-versions” of measurable cardinals, and indeed, a result of Kunen shows that Ramsey cardinals fall into this category as well. In this case, given any , we can find as above such that and that the -measure induced by is furthermore countably complete, meaning that intersections of a countable sequences (in ) of elements of are nonempty, and weakly amenable, meaning for every having -cardinality .
If we remove these conditions of countable completeness and weak amenability then we arrive at an equivalent characterisation of weakly compact cardinals, and Gitman (2011) introduced strengthenings as well called strongly Ramsey cardinals and super Ramsey cardinals. Holy and Schlicht (2018) further added onto the catalogue by introducing a whole hierarchy of -Ramsey cardinals for being a cardinal. Their definition is based upon the following game of length .
Here all the ‘s are weak -models and the ‘s are -measures. We require that the models are all elementary in some large , that the models and measures are -increasing and that, letting be the union of all the measures and the union of all the models, is required to be an -normal -measure with a well-founded ultrapower.
We then say that a cardinal is -Ramsey if player I doesn’t have a winning strategy in the above game (the game doesn’t depend upon ). So this is a kind of “filter extension property”. We can also define the ostensibly stronger notion of being strategic -Ramsey if player II does have a winning strategy.
In our paper we firstly tweak the game to be able to define (strategic) -Ramsey cardinals for every ordinal , and proceed to show a bunch of properties these cardinals satisfy, some of which are mentioned in the abstract. In particular, we showed the following theorem, which is Theorem 5.17 and 5.19 in the paper.
Theorem (N., Welch). Let be strategic -Ramsey. Then
- Either there is a sharp for a strong cardinal, or is measurable in the core model , and
- There is a precipitous ideal on .
This in particular shows that strategic -Ramseys are equiconsistent with the existence of a measurable cardinal (in two different ways), so that these Ramsey-like cardinals can reach the measurable cardinals. This makes it likely that cardinals between weakly compacts and measurables can be characterised by some kind of Ramsey-like cardinal. There’s also the question of whether the game that we’re considering is determined, which would mean that the -Ramsey cardinals are equivalent to their strategic counterparts. The above theorem shows that this is not true when is uncountable, as it’s known from Holy and Schlicht (2018) that measurables are consistency-wise stronger than -Ramseys for all . We show the following, which is Theorem 4.10 in the paper.
Theorem (N., Welch). Every -Ramsey cardinal is strategic -Ramsey.
This then leaves open whether the games are determined for countable , which might be connected to Neeman’s theory of long games. All the large cardinals considered in the paper are here depicted, one with the consistency-wise implications and the other with the direct implications. These diagrams can also be found in high-quality pdfs here.
- Feng, Q. (1990). A hierarchy of Ramsey cardinals. Annals of Pure and Applied Logic, 49(2):257–277.
- Gitman, V. (2011). Ramsey-like cardinals. The Journal of Symbolic Logic, 76(2):519–540.
- Holy, P. and Schlicht, P. (2018). A hierarchy of Ramsey-like cardinals. Fundamenta Mathematicae. DOI: 10.4064/fm396-9-2017.
- Sharpe, I. and Welch, P. D. (2011). Greatly Erdős cardinals with some generalizations to the Chang and Ramsey properties. Annals of Pure and Applied Logic, 162:863–902.