Projektdetails
Abstract
Wider research context/theoretical framework
Two important areas of set theory are the structure of classes of uncountable combinatorial objects such as trees and linear orders and the possible structure of the real line including the study of cardinal characteristics and descriptive set theory. Both of these subfields have contributed significantly to the applicability of set theory but so far they have had for the most part only the most perfunctory interaction.
The overall objective of the proposed project therefore is to apply ideas from set theory of the reals to the study of uncountable objects and visa versa with a particular eye towards how cardinal characteristics may affect the existence of combinatorial objects with certain properties.
Hypotheses/research questions/objectives
The research questions of the proposal break down into four related areas.
1. Maximal Sets
In this part we will consider consistent inequalities between cardinal characteristics associated with maximal sets of reals such as MAD families and their relatives. The forcing innovations will come in part from recent work by the PI and his collaborators on new applications of filters to such sets. One problem of particular interest here is the consistency of a greater than other cardinals associated to maximal sets.
2. Trees
Here we will look into how the real line affects the structure of generalized Aronszajn trees, building on work of the author computing cardinal characteristics in the special tree model. We are particularly interested in the question of whether it is consistent that every Aronszajn tree of size d is special.
3. Linear Orders
Here we will compare cardinal characteristics to sets of reals as linear orders. The guiding question here is that of Todorcevic: is BA(p) consistent?
4. Posets and Forcing Notions
This last section focuses on forcing. It involves building new partials orders which can be used to prove the above consistencies.
Approach/Methods
The approach to the problems described above has two prongs. First we will look at ZFC combinatorics to prove relations between the objects under consideration. Then, we will use the technique of forcing, particularly novel methods like template iterations and forcing with side conditions to provide models witnessing consistency results.
Level of originality/innovation
The level of originality and innovation in the proposed project will be high. Currently there has been minimal interaction between combinatorics of uncountable structures and set theory of the reals. As such it is anticipated that new techniques in forcing and combinatorics will be needed.
Primary Researchers involved
The primary researchers will be the PI, Dr. Corey Bacal Switzer and the mentor, Ass-Prof Dr. Vera Fischer.
Two important areas of set theory are the structure of classes of uncountable combinatorial objects such as trees and linear orders and the possible structure of the real line including the study of cardinal characteristics and descriptive set theory. Both of these subfields have contributed significantly to the applicability of set theory but so far they have had for the most part only the most perfunctory interaction.
The overall objective of the proposed project therefore is to apply ideas from set theory of the reals to the study of uncountable objects and visa versa with a particular eye towards how cardinal characteristics may affect the existence of combinatorial objects with certain properties.
Hypotheses/research questions/objectives
The research questions of the proposal break down into four related areas.
1. Maximal Sets
In this part we will consider consistent inequalities between cardinal characteristics associated with maximal sets of reals such as MAD families and their relatives. The forcing innovations will come in part from recent work by the PI and his collaborators on new applications of filters to such sets. One problem of particular interest here is the consistency of a greater than other cardinals associated to maximal sets.
2. Trees
Here we will look into how the real line affects the structure of generalized Aronszajn trees, building on work of the author computing cardinal characteristics in the special tree model. We are particularly interested in the question of whether it is consistent that every Aronszajn tree of size d is special.
3. Linear Orders
Here we will compare cardinal characteristics to sets of reals as linear orders. The guiding question here is that of Todorcevic: is BA(p) consistent?
4. Posets and Forcing Notions
This last section focuses on forcing. It involves building new partials orders which can be used to prove the above consistencies.
Approach/Methods
The approach to the problems described above has two prongs. First we will look at ZFC combinatorics to prove relations between the objects under consideration. Then, we will use the technique of forcing, particularly novel methods like template iterations and forcing with side conditions to provide models witnessing consistency results.
Level of originality/innovation
The level of originality and innovation in the proposed project will be high. Currently there has been minimal interaction between combinatorics of uncountable structures and set theory of the reals. As such it is anticipated that new techniques in forcing and combinatorics will be needed.
Primary Researchers involved
The primary researchers will be the PI, Dr. Corey Bacal Switzer and the mentor, Ass-Prof Dr. Vera Fischer.
| Kurztitel | Mengenlehre-reelle Zahlen u Kombinatorik |
|---|---|
| Status | Laufend |
| Tatsächlicher Beginn/ -es Ende | 1/11/23 → 31/10/26 |