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Strong Axioms of Infinity Frameworks Interactions and Applications

In spite of their central role in modern set theory, strong axioms of infinity (or large cardinal axioms) are still surrounded by an aura of vagueness, a lack of generality and many open conceptual questions. After the study of la... ver más

31/07/2022

173K€

Presupuesto del proyecto: 173K€

Líder del proyecto

UNIVERSIDAD DE BARCELONA No se ha especificado una descripción o un objeto social para esta compañía.

Total investigadores 328

Fecha límite participación Sin fecha límite de participación.

Financiación concedida El organismo H2020 notifico la concesión del proyecto el día 2022-07-31

MSCA-IF-2018: Individual Fellowships

I+D

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1 Participantes

172.93K€ | Lider

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Duración del proyecto: 39 meses Fecha Inicio: 2019-04-23
Fecha Fin: 2022-07-31

Líder del proyecto

UNIVERSIDAD DE BARCELONA No se ha especificado una descripción o un objeto social para esta compañía.

Total investigadores 328

Presupuesto del proyecto 173K€

Fecha límite de participación Sin fecha límite de participación.

Descripción del proyecto In spite of their central role in modern set theory, strong axioms of infinity (or large cardinal axioms) are still surrounded by an aura of vagueness, a lack of generality and many open conceptual questions. After the study of large cardinals has evolved for over eighty years, recent results suggest that it now makes sense to develop a general theory of strong axioms of infinity in which all known large cardinals are seen as milestones in a hierarchy of mathematical principles derived from some much more general considerations about the reflective properties of the set-theoretic universe. The development of such a theory would lead to a breakthrough in our understanding of large cardinals and their role in mathematics, and provide strong justifications for their acceptance as true mathematical statements. In this project, we want to work towards this breakthrough with the help of novel combinations of concepts and techniques from different areas of set theory. We will develop general frameworks for strong axioms of infinity that incorporate all types of large cardinals studied so far. The work of the proposed supervisor on structural reflection properties and recent pioneering results in combinatorial set theory will serve as the starting points for this work. Moreover, motivated by the strong influence of large cardinals on the theory of definable sets of real numbers, we will study the impact of these axioms on definability at higher cardinalities. This task is closely related to one of the most important developments in modern set theory, Hugh Woodin’s programme of constructing a canonical inner model containing a supercompact cardinal. Finally, strong axioms of infinity have recently been used with great success to answer questions in other branches of mathematics, like category theory or homotopy theory. These results opened up a wide area of possible applications of set-theoretic results that we also want to explore in our project.

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