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Set theory beyond the first uncountable cardinal

Total Cost €


EC-Contrib. €






Project "BeyondA1" data sheet

The following table provides information about the project.


Organization address
postcode: 52900

contact info
title: n.a.
name: n.a.
surname: n.a.
function: n.a.
email: n.a.
telephone: n.a.
fax: n.a.

 Coordinator Country Israel [IL]
 Total cost 1˙362˙500 €
 EC max contribution 1˙362˙500 € (100%)
 Programme 1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
 Code Call ERC-2018-STG
 Funding Scheme ERC-STG
 Starting year 2018
 Duration (year-month-day) from 2018-10-01   to  2023-09-30


Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 
1    BAR ILAN UNIVERSITY IL (RAMAT GAN) coordinator 1˙362˙500.00


 Project objective

We propose to establish a research group that will unveil the combinatorial nature of the second uncountable cardinal. This includes its Ramsey-theoretic, order-theoretic, graph-theoretic and topological features. Among others, we will be directly addressing fundamental problems due to Erdos, Rado, Galvin, and Shelah. While some of these problems are old and well-known, an unexpected series of breakthroughs from the last three years suggest that now is a promising point in time to carry out such a project. Indeed, through a short period, four previously unattainable problems concerning the second uncountable cardinal were successfully tackled: Aspero on a club-guessing problem of Shelah, Krueger on the club-isomorphism problem for Aronszajn trees, Neeman on the isomorphism problem for dense sets of reals, and the PI on the Souslin problem. Each of these results was obtained through the development of a completely new technical framework, and these frameworks could now pave the way for the solution of some major open questions. A goal of the highest risk in this project is the discovery of a consistent (possibly, parameterized) forcing axiom that will (preferably, simultaneously) provide structure theorems for stationary sets, linearly ordered sets, trees, graphs, and partition relations, as well as the refutation of various forms of club-guessing principles, all at the level of the second uncountable cardinal. In comparison, at the level of the first uncountable cardinal, a forcing axiom due to Foreman, Magidor and Shelah achieves exactly that. To approach our goals, the proposed project is divided into four core areas: Uncountable trees, Ramsey theory on ordinals, Club-guessing principles, and Forcing Axioms. There is a rich bilateral interaction between any pair of the four different cores, but the proposed division will allow an efficient allocation of manpower, and will increase the chances of parallel success.


year authors and title journal last update
List of publications.
2019 Ari Meir Brodsky, Assaf Rinot
A Remark on Schimmerling’s Question
published pages: 525-561, ISSN: 0167-8094, DOI: 10.1007/s11083-019-09482-7
Order 36/3 2020-03-23
2019 Assaf Rinot
Souslin trees at successors of regular cardinals
published pages: 200-204, ISSN: 0942-5616, DOI: 10.1002/malq.201800065
Mathematical Logic Quarterly 65/2 2020-03-23

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