11 research outputs found
A hierarchy of Ramsey-like cardinals
We introduce a hierarchy of large cardinals between weakly compact and
measurable cardinals, that is closely related to the Ramsey-like cardinals
introduced by Victoria Gitman, and is based on certain infinite filter games,
however also has a range of equivalent characterizations in terms of elementary
embeddings. The aim of this paper is to locate the Ramsey-like cardinals
studied by Gitman, and other well-known large cardinal notions, in this
hierarchy
Two-cardinal ideal operators and indescribability
A well-known version of Rowbottom's theorem for supercompactness ultrafilters
leads naturally to notions of two-cardinal Ramseyness and corresponding normal
ideals introduced herein. Generalizing results of Baumgartner [7, 8], Feng [22]
and the first author [16, 17], we study the hierarchies associated with a
particular version of two-cardinal Ramseyness and a strong version of
two-cardinal ineffability, as well as the relationships between these
hierarchies and a natural notion of transfinite two-cardinal indescribability.Comment: Fixed more typos in the first paragrap
Games and Ramsey-like cardinals
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).Comment: 33 pages, 2 figures. Added Theorem 4.20 saying that strategic
-Ramsey cardinals are equiconsistent with measurables, and
fixed many typos. This version is forthcoming in the JS
Ideal operators and higher indescribability
We investigate properties of the ineffability and the Ramsey operator, and a
common generalization of those that was introduced by the second author, with
respect to higher indescribability, as introduced by the first author. This
extends earlier investigations on the ineffability operator by James
Baumgartner, and on the Ramsey operator by Qi Feng, by Philip Welch et al. and
by the first author.Comment: Fixed minor typos and error
Chinese Research on Mathematical Logic and the Foundations of Mathematics
This paper outlines the Chinese research on mathematical logic and the foundations of mathematics. Firstly, it presents the introduction and spread of mathematical logic in China, especially the teaching and translation of mathematical logic initiated by Bertrand Russell’s lectures in the country. Secondly, it outlines the Chinese research on mathematical logic after the founding of the People’s Republic of China. The research in this period experienced a short revival under the criticism of the Soviet Union, explorations under the heavy influence of the Cultural Revolution, and the vigorous development of mathematical logic teaching and research after the period of “Reform and Opening Up” that started in the late 1970s, and the full integration of Chinese mathematical logic research into the international academic circle in the new century after 2000. In the third part, it focuses on the unique and original results of the Chinese mathematical logic research teams from the following three aspects: medium logic, lattice implication algebras and their lattice-valued systems of logic, and Chinese notation of logical constants, which can be used as a substantive supplement to the relevant literature on the history of mathematical logic in China. The last part is a reflection on the shortcomings of contemporary Chinese research on mathematical logic and the foundations of mathematics