The continuum, forcing and large cardinals
The continuum, forcing and large cardinals
Bilaterale Ausschreibung: Tschechien
Disciplines
Mathematics (100%)
Keywords
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Tree property,
Singular Cardinals,
Mathias forcing,
Selection Principles,
Template Iterations
The work shall study interactions between large cardinals and forcing and also with regard to properties of the continuum. The work shall be divided into three subtopics: (1) First subtopic is called "Combinatorial properties of cardinals and the continuum function". The focus shall be on the impact of several combinatorial properties on the continuum function (a function which maps kappa to the size of the powerset of kappa). As Easton showed, without extra assumptions, there is very little ZFC can prove about the continuum function on regulars. However, once we start to consider large cardinals, or in general combinatorial properties often formulated in the context of large cardinals (tree property, square principles, etc.), suddenly the situation is much more interesting. Subtopic 1 studies these connections. (2) Second subtopic is called "Combinatorial characteristics of the continuum and the Mathias forcing". This subtopic focuses on the real line from the point of certain combinatorial characteristics which shall be studied by means of the Mathias forcing. This forcing is determined by a filter on the powerset of omega. Thus it is natural to study the connections between the combinatorial properties of filters and the properties of the real line. This subtopic combines set theory and topology. (3) The third subtopic is called "New developments in the template iterations." Template iterations is a modern forcing method with the potential to show new results regarding the combinatorial characteristics of the real line. Among the main problems to be studied are: obtaining cardinal characteristics with singular values, application of template iterations on non-definable posets, and finally the application of the method with the mixed finite and countable support. These new approaches will be useful in obtaining new results.
This project explored several aspects of set theory, the study of infinity. We explored the behaviour of large infinities (also known as large cardinals), the combinatorial properties of orderings and graphs, as well as specific properties of the continuum (the real line). The project was carried out jointly by the Vienna group at the Kurt Goedel Research Center and the Prague group at the Logic Department of Charles University.
- Universität Wien - 100%
- Radek Honzik, Charles University Prague - Czechia
Research Output
- 74 Citations
- 46 Publications
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2025
Title Good projective witnesses DOI 10.1016/j.apal.2025.103606 Type Journal Article Author Fischer V Journal Annals of Pure and Applied Logic Pages 103606 Link Publication -
2019
Title A model with Suslin trees but no minimal uncountable linear orders other than ?1 and -?1 DOI 10.1007/s11856-019-1899-x Type Journal Article Author Soukup D Journal Israel Journal of Mathematics Pages 199-224 -
2019
Title Infinite monochromatic sumsets for colourings of the reals DOI 10.1090/proc/14431 Type Journal Article Author Komjáth P Journal Proceedings of the American Mathematical Society Pages 2673-2684 Link Publication -
2019
Title Cycle reversions and dichromatic number in tournaments DOI 10.1016/j.ejc.2018.10.008 Type Journal Article Author Ellis P Journal European Journal of Combinatorics Pages 31-48 Link Publication -
2019
Title Balanced independent sets in graphs omitting large cliques DOI 10.1016/j.jctb.2018.11.006 Type Journal Article Author Laflamme C Journal Journal of Combinatorial Theory, Series B Pages 1-9 Link Publication -
2018
Title On Spaces with Sigma-Closed-Discrete Dense Sets Type Journal Article Author Dias R.R. Journal Topology Proceedings Pages 245-264 Link Publication -
2018
Title The open dihypergraph dichotomy and the second level of the Borel hierarchy DOI 10.48550/arxiv.1803.03205 Type Preprint Author Carroy R -
2018
Title Two infinite quantities and their surprising relationship DOI 10.48550/arxiv.1803.04331 Type Preprint Author Soukup D -
2018
Title A model with Suslin trees but no minimal uncountable linear orders other than $\omega_1$ and $-\omega_1$ DOI 10.48550/arxiv.1803.03583 Type Preprint Author Soukup D -
2018
Title The tree property at the double successor of a singular cardinal with a larger gap DOI 10.1016/j.apal.2018.02.002 Type Journal Article Author Friedman S Journal Annals of Pure and Applied Logic Pages 548-564 -
2018
Title Uncountable Strongly Surjective Linear Orders DOI 10.1007/s11083-018-9454-7 Type Journal Article Author Soukup D Journal Order Pages 43-64 Link Publication -
2018
Title The tree property and the continuum function below DOI 10.1002/malq.201600028 Type Journal Article Author Honzik R Journal Mathematical Logic Quarterly Pages 89-102 -
2018
Title More ZFC inequalities between cardinal invariants DOI 10.48550/arxiv.1802.02791 Type Preprint Author Fischer V -
2018
Title TOWERS IN FILTERS, CARDINAL INVARIANTS, AND LUZIN TYPE FAMILIES DOI 10.1017/jsl.2017.52 Type Journal Article Author Brendle J Journal The Journal of Symbolic Logic Pages 1013-1062 Link Publication -
2018
Title INFINITE COMBINATORICS PLAIN AND SIMPLE DOI 10.1017/jsl.2018.8 Type Journal Article Author Soukup D Journal The Journal of Symbolic Logic Pages 1247-1281 Link Publication -
2022
Title Ideal topologies in higher descriptive set theory DOI 10.1016/j.apal.2021.103061 Type Journal Article Author Holy P Journal Annals of Pure and Applied Logic Pages 103061 Link Publication -
2021
Title On the complexity of classes of uncountable structures: trees on $\aleph _1$ DOI 10.4064/fm910-6-2020 Type Journal Article Author Friedman S Journal Fundamenta Mathematicae Pages 175-196 Link Publication -
2019
Title Reducing the dichromatic number via cycle reversions in infinite digraphs DOI 10.48550/arxiv.1909.00873 Type Preprint Author Ellis P -
2019
Title A 0-dimensional, Lindelöf space that is not strongly D DOI 10.48550/arxiv.1902.06500 Type Preprint Author Soukup D -
2019
Title On the complexity of classes of uncountable structures: trees on $\aleph_1$ DOI 10.48550/arxiv.1906.00849 Type Preprint Author Friedman S -
2019
Title A 0-dimensional, Lindelöf space that is not strongly D DOI 10.1016/j.topol.2019.106832 Type Journal Article Author Soukup D Journal Topology and its Applications Pages 106832 Link Publication -
2020
Title Reducing the dichromatic number via cycle reversions in infinite digraphs DOI 10.1016/j.ejc.2020.103196 Type Journal Article Author Ellis P Journal European Journal of Combinatorics Pages 103196 Link Publication -
2020
Title Laver Trees in the Generalized Baire Space DOI 10.48550/arxiv.2009.01886 Type Preprint Author Khomskii Y -
2016
Title Towers in filters, cardinal invariants, and Luzin type families DOI 10.48550/arxiv.1605.04735 Type Preprint Author Brendle J -
2016
Title $G_\delta $ and co-meager semifilters DOI 10.4064/fm182-2-2016 Type Journal Article Author Brian W Journal Fundamenta Mathematicae Pages 1-14 Link Publication -
2017
Title Infinite combinatorics plain and simple DOI 10.48550/arxiv.1705.06195 Type Preprint Author Soukup D -
2017
Title Infinite monochromatic sumsets for colourings of the reals DOI 10.48550/arxiv.1710.07500 Type Preprint Author Komjáth P -
2017
Title Uncountable strongly surjective linear orders DOI 10.48550/arxiv.1706.10171 Type Preprint Author Soukup D -
2020
Title The open dihypergraph dichotomy and the second level of the Borel hierarchy DOI 10.1090/conm/752/15127 Type Book Chapter Author Carroy R Publisher American Mathematical Society (AMS) Pages 1-19 Link Publication -
2015
Title Decompositions of edge-coloured infinite complete graphs into monochromatic paths II DOI 10.48550/arxiv.1507.06187 Type Preprint Author Soukup D -
2017
Title Grigorieff forcing and the tree property DOI 10.14712/24647055.2017.16 Type Journal Article Author Stejskalová Š Journal AUC PHILOSOPHICA ET HISTORICA Pages 63-72 Link Publication -
2017
Title Preserving measurability with Cohen iterations DOI 10.14712/24647055.2017.13 Type Journal Article Author Honzík R Journal AUC PHILOSOPHICA ET HISTORICA Pages 27-32 Link Publication -
2017
Title Cycle reversions and dichromatic number in tournaments DOI 10.48550/arxiv.1708.02441 Type Preprint Author Ellis P -
2017
Title On spaces with $\sigma$-closed-discrete dense sets DOI 10.48550/arxiv.1701.00356 Type Preprint Author Dias R -
2017
Title Decompositions of edge-colored infinite complete graphs into monochromatic paths DOI 10.1016/j.disc.2016.09.028 Type Journal Article Author Elekes M Journal Discrete Mathematics Pages 2053-2069 Link Publication -
0
Title The tree property at alphe_omega+2 with aleph_omega strong limit and 2^aleph_omega large Type Journal Article Author Friedman S.-D. Journal Fundamenta Mathematicae Pages 1-26 Link Publication -
0
Title 0022-4812 Type Journal Article Author Fischer V. Journal Journal of Symbolic Logic Pages 1-15 Link Publication -
0
Title The open dihypergraph dichotomy and the second level of the Borel hierarchy1 Type Journal Article Author Carroy R. Journal Contemporary mathematics : CONM Pages 1-24 Link Publication -
2022
Title Laver trees in the generalized Baire space DOI 10.1007/s11856-022-2465-5 Type Journal Article Author Khomskii Y Journal Israel Journal of Mathematics Pages 599-620 Link Publication -
2021
Title MORE ZFC INEQUALITIES BETWEEN CARDINAL INVARIANTS DOI 10.1017/jsl.2021.52 Type Journal Article Author Fischer V Journal The Journal of Symbolic Logic Pages 897-912 Link Publication -
2021
Title A universal partition result for infinite homogeneous K n -free and related graphs DOI 10.1016/j.disc.2020.112153 Type Journal Article Author Aranda A Journal Discrete Mathematics Pages 112153 Link Publication -
2021
Title Ideal Topologies in Higher Descriptive Set Theory DOI 10.48550/arxiv.2111.07339 Type Preprint Author Holy P -
2018
Title A Laver-like indestructibility for hypermeasurable cardinals DOI 10.1007/s00153-018-0637-0 Type Journal Article Author Honzik R Journal Archive for Mathematical Logic Pages 275-287 -
2018
Title Ladder system uniformization on trees I & II DOI 10.48550/arxiv.1806.03867 Type Preprint Author Soukup D -
2017
Title Decompositions of edge-coloured infinite complete graphs into monochromatic paths II DOI 10.1007/s11856-017-1552-5 Type Journal Article Author Soukup D Journal Israel Journal of Mathematics Pages 235-273 -
2017
Title Orientations of graphs with uncountable chromatic number DOI 10.1002/jgt.22233 Type Journal Article Author Soukup D Journal Journal of Graph Theory Pages 606-630 Link Publication