REGPROP · Regularity properties, definability and combinatorics on the real line.
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2017-03-01 → 2019-02-28
- EU contribution
- €171,461
- Participants
- 1
- Scheme
- MSCA-IF-EF-ST
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Results in brief
Regularity properties, definability and combinatorics on the real line.
This research project is in set theory, an area of abstract mathematics dealing with foundational questions such as "what is the nature of the real number continuum"', "what can be formally proved on the basis of commonly accepted axioms", and "what are the limitations of mathematical formalism". The topic of this research project was regularity properties of sets of real numbers. To give an example: Lebesgue measure is the standard mathematical formalism to capture the intuition of "size" or "volume" of an object in space; a straightforward definition assuming that all sets can be measured in volume, however, leads to paradoxes (such as the famous Banach-Tarski paradox). As a consequence, we have to accept that not all sets are Lebesgue measurable: some sets are and others are not. We call Lebesgue measurability a regularity property since it separates the regular from the irregular sets. Usually, sets with simple definitions are regular while sets with complicated definitions can be irregular. One of the goals of set theory is to understand the extent of regularity in the universe of sets and characterise exactly what the conditions for regularity and irregularity as well as the interplay between different regularity properties is. The study of regularity properties is a well-established subfield of set theory to which this project contributed by providing an extension of the analysis to more complicated sets, providing new examples of regularity properties, and developing new techniques for their study.
Data: CORDIS, © European Union
Project objective
This proposed research is in mathematical logic and foundations of mathematics, more specifically in set theory. It is motivated by the interplay between regularity properties and definability for subsets of real numbers. By ""regularity properties"" we are referring to certain desirable properties of sets, and by ""definability"" to the logical description of such sets, in the sense of Descriptive Set Theory.The study of such questions goes back to classical issues in topology, analysis and related fields of mathematics, raised by the great pioneers of abstract mathematics of the late 19th and early 20th century, such as Georg Cantor, Emile Borel, Henri Lebesgue and others. These mathematicians were faced with seemingly insurmountable challenges which could only be resolved later with the advent of logical and meta-mathematical methods, developed by Kurt Gödel in 1938 and by Paul Cohen in 1964.Since that time, the study of Regularity Properties has continued to hold a central position in the foundations of mathematics. Many mathematicians and logicians of high status and prestige have contributed to this area, among them W. Hugh Woodin (winner of the Hausdorff Medal 2013), Stevo Todorčević (winner of the CRM-Fields-PIMS prize 2012) and Saharon Shelah (winner of numerous awards, among them the Erdös Prize 1977 and the Karp Prize 1983).We propose to contribute to this line of research in a number of interrelated directions, such as: studying new regularity properties (relevant to other fields of mathematics), developing abstract frameworks for such properties, studying higher complexity classes, and generalising results to spaces other than the classical real numbers. Several technical results involving the method of ""forcing"" needed to construct models of set theory, will also be worked out along the way.""
Original text from CORDIS.
Participants
- UNIVERSITY OF HAMBURG · HamburgCoordinatorGermany
Links
Data: CORDIS, © European Union