Forcing Methods: Creatures, Products and Iterations
Forcing Methods: Creatures, Products and Iterations
Disciplines
Mathematics (100%)
Keywords
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Creature Forcing,
Iterated Forcing,
Set Theory Of The Reals,
Combinatorial Set Theory,
Cardinal Characteristics
The focus of this project is research in set theory, a subfield of mathematical logic. In set theory we consider cardinalities, which measure the size of infinite sets. We are in particular interested in the set of all real numbers, and in subsets of this set. Many questions about properties of these sets (e.g. in connection with the Lebesgue measure) cannot be answered with the usual set-theoretic axioms. The method of forcing allows us to construct set-theoretic universes in which these answers can be depending on what you want yes or no. In this project we try to further develop the method of forcing. On the one hand, to construct new set- theoretic universes, and on the other hand, in order to better understand the basic properties of the set of real numbers and its subsets.
In this project we have investigated several infinities (or or "infinite cardinal numbers"), which appear as cardinalities of sets of real numbers. Set Theory, a subfield of Mathematical Logic deals with the investigation of infinite sets (whereas in everyday life we mainly deal with finite objects); examples for such sets are the set of all real numbers, or its subsets such as the set of natural numbers, or the set of numbers between 0 and 1. Not only finite sets but also infinite sets can be compared in terms of their "size" (we use the technical term "cardinality", to point out that this is different from the notion of size of a finite set), and it has long been known that there are infinitely many cardinalities of infinite sets; for example the set of natural numbers has the same cardinality as the set of all rational numbers, but not the same cardinality as the set of all real numbers. What exactly the difference between these two cardinalities is has long been a central question in set theory; Georg Cantor`s Continuum Hypothesis, the first of Hilbert`s 23 problems), was the conjecture that no cardinality lies strictly between those two.In the project we have investigated cardinalities of special (often pathological) sets which appear in other areas of mathematics. For example: What is the least cardinality of a subset of 3-dimensional space to which one cannot assign a "volume" in a meaningful way. (Such sets appear in the Banach-Tarski paradox.) When we consider, for example, subsets of 3-dimensional space, we distinguish "null sets" (i.e., sets to which we assign the volume 0), for example finite subsets or 2-dimensional subsets such as planes) and "positive" sets (=all other sets). We are interested, for example, in the smallest cardinality of a positive set, in the number of null sets needed whose join will the whole space. The two cardinalities just described, together with 8 more, are collected in Cichon`s diagram, which also describes the relations between them. They all are uncountable (that is, larger than the cardinality of the set of all natural numbers). But one can construct a set theoretic universe in which all these cardinalities are equal, and further universes where some of them have distinct values. By combining known methods with new ideas, we could (for the first time) describe a set theoretic universe in which all these cardinalities have different values.Our new method has also found other applications. There is a long list of further cardinalities, whose relation to the cardinalities considered so far is still open.
- Technische Universität Wien - 100%
Research Output
- 26 Citations
- 28 Publications
- 2 Scientific Awards
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2020
Title Stranger things about forcing without AC DOI 10.14712/1213-7243.2020.013 Type Journal Article Author Martin G Journal Commentationes Mathematicae Universitatis Carolinae Pages 21-26 Link Publication -
2021
Title Preservation of splitting families and cardinal characteristics of the continuum DOI 10.1007/s11856-021-2237-7 Type Journal Article Author Goldstern M Journal Israel Journal of Mathematics Pages 73-129 Link Publication -
2021
Title Many different uniformity numbers of Yorioka ideals DOI 10.1007/s00153-021-00809-z Type Journal Article Author Klausner L Journal Archive for Mathematical Logic Pages 653-683 Link Publication -
2021
Title Cichon’s maximum without large cardinals DOI 10.4171/jems/1178 Type Journal Article Author Goldstern M Journal Journal of the European Mathematical Society Pages 3951-3967 Link Publication -
2020
Title THE POLARISED PARTITION RELATION FOR ORDER TYPES DOI 10.1093/qmathj/haaa003 Type Journal Article Author Klausner L Journal The Quarterly Journal of Mathematics Pages 823-842 Link Publication -
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Title Cichon's maximum without large cardinals Type Journal Article Author Goldstern M Journal Journal of the European Mathematical Society -
2021
Title Preservation of splitting families and cardinal characteristics of the continuum Type Journal Article Author Goldstern M Journal Israel J. Math. Pages 73-129 -
0
Title preservation of splitting families and cardinal characteristics of the continuum Type Other Author Goldstern M -
2021
Title Controlling cardinal characteristics without adding reals Type Journal Article Author Goldstern M Journal J. Math. Log. -
0
Title controlling cardinal characteristics without adding reals Type Other Author Goldstern M -
0
Title Cichoń's diagram and localisation cardinals Type Journal Article Author Goldstern M Journal Archive for Mathematical Logic -
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Title Cichoń's diagram and localisation cardinals Type Journal Article Author Goldstern M Journal Archive for Mathematical Logic -
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Title The Higher Cichon Diagram Type Journal Article Author Baumhauer T Journal Fundamenta Mathematicae -
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Title The Higher Cichon Diagram Type Journal Article Author Baumhauer T Journal Fundamenta Mathematicae -
2019
Title Cichon's Maximum. Type Journal Article Author Goldstern M Journal Annals of Mathematics Pages 113-143 -
2022
Title Controlling classical cardinal characteristics while collapsing cardinals DOI 10.4064/cm8420-2-2022 Type Journal Article Author Shelah S Journal Colloquium Mathematicum -
2021
Title The higher Cichoń diagram DOI 10.4064/fm666-4-2020 Type Journal Article Author Shelah S Journal Fundamenta Mathematicae -
2018
Title The Higher Cichoń Diagram DOI 10.48550/arxiv.1806.08583 Type Other Author Baumhauer T -
2018
Title Halfway New Cardinal Characteristics DOI 10.48550/arxiv.1808.02442 Type Other Author Brendle J -
2019
Title Controlling classical cardinal characteristics while collapsing cardinals DOI 10.48550/arxiv.1904.02617 Type Preprint Author Goldstern M -
2020
Title Preservation of splitting families and cardinal characteristics of the continuum DOI 10.48550/arxiv.2007.13500 Type Preprint Author Goldstern M -
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Title Two Simple Facts about Non-AC Forcing. Type Other Author Goldstern M -
2023
Title Halfway new cardinal characteristics DOI 10.1016/j.apal.2023.103303 Type Journal Article Author Brendle J Journal Annals of Pure and Applied Logic Pages 103303 Link Publication -
2020
Title Cichon’s diagram and localisation cardinals DOI 10.1007/s00153-020-00746-3 Type Journal Article Author Goldstern M Journal Archive for Mathematical Logic Pages 343-411 Link Publication -
2020
Title Controlling cardinal characteristics without adding reals DOI 10.1142/s0219061321500185 Type Journal Article Author Goldstern M Journal Journal of Mathematical Logic Pages 2150018 Link Publication -
2019
Title Set-theoretic blockchains DOI 10.1007/s00153-019-00672-z Type Journal Article Author Habic M Journal Archive for Mathematical Logic Pages 965-997 Link Publication -
2019
Title Cichon's maximum DOI 10.4007/annals.2019.190.1.2 Type Journal Article Author Goldstern M Journal Annals of Mathematics Link Publication -
2019
Title Another ordering of the ten cardinal characteristics in Cichon's diagram DOI 10.14712/1213-7243.2015.273 Type Journal Article Author Jakob K Journal Commentationes Mathematicae Universitatis Carolinae Pages 61-95 Link Publication
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2019
Title Banff Set theory of the Reals workshop, Oaxaca Type Personally asked as a key note speaker to a conference Level of Recognition Continental/International -
2019
Title Young Set Theory, Novi Sad Type Personally asked as a key note speaker to a conference Level of Recognition Continental/International