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Large Properties at Small Cardinals

Large cardinals are deeply used in set theory as many problems that are independent from the classical theory ZFC can be solved under the assumption that large cardinals exist. One of the most rich and fruitful research areas in c... ver más

31/05/2016

THE HEBREW UNIVERS...

192K€

Presupuesto del proyecto: 192K€

Líder del proyecto

THE HEBREW UNIVERSITY OF JERUSALEM No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

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Financiación concedida El organismo FP7 notifico la concesión del proyecto el día 2016-05-31 No tenemos la información de la convocatoria

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Líder del proyecto

THE HEBREW UNIVERSITY OF JERUSALEM No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Presupuesto del proyecto 192K€

Fecha límite de participación Sin fecha límite de participación.

Descripción del proyecto Large cardinals are deeply used in set theory as many problems that are independent from the classical theory ZFC can be solved under the assumption that large cardinals exist. One of the most rich and fruitful research areas in contemporary set theory is the study of those properties of large cardinals that can hold even at small cardinals. Two properties of that kind are particularly interesting, the strong tree property and the super tree property. These properties provide a simple combinatorial characterisations of very strong large cardinal notions, namely strongly compact cardinals and supercompact cardinals. Although the strong and super tree property are associated with such strong large cardinals, they can be satisfied even by small cardinals such as aleph_2. The project focuses on some potential applications of such properties at small cardinals. Two conjectures are stated, the first one implies that the strong tree property has strong consequences in the arithmetic of infinite cardinals, namely we conjecture that the strong tree property at aleph_2 implies the singular cardinal hypothesis. The second conjecture concerns the study of infinite almost abelian groups, more precisely we ask whether the super tree property at a small cardinal kappa implies that every almost free abelian group of size kappa is free. The project is concerned more generally with developing combinatorial characterisations of all large cardinal notions and it proposed a systematic analysis of the combinatorial principles associated with large cardinals.

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