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Results tagged with descriptive-set-theory
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Descriptive Set Theory is the study of definable subsets of Polish spaces, where definable is taken to mean from the Borel or projective hierarchies. Other topics include infinite games and determinacy, definable equivalence relations and Borel reductions between them, Polish groups, and effective descriptive set theory.
3
votes
Related Open Game in Analytic Determinacy
One possible space is the space of all sequences $(a_0,(b_0,h_0),a_1,(b_1,h_1),\dots)$,
where the $a_i$ and $b_i$ are natural numbers and the $h_i$ are order preserving maps
from a finite subset of $S …
- 16.2k
4
votes
Accepted
Does recent work of Woodin clash with an older result in Descriptive Set Theory?
What is the problem? Large cardinals are consistent with CH.
This does not require looking at Ultimate L. But large cardinals are also
consistent with failures of CH. And if you are in a model whe …
- 16.2k
18
votes
Definable Wellordering of the Reals
Several objects can be defined from a wellordering of the reals. Nonprincipal ultrafilters on
the natural numbers, non-measurable sets of reals, sets of reals without the property of Baire, and so on …
- 16.2k
11
votes
Accepted
Decomposing $\mathbf{\Pi}^1_1$ sets into closed sets
Every ${\bf\Pi}_1^1$ set is the union of $\aleph_1$ Borel sets, so we only have to make sure that every Borel set is the union of $\aleph_1$ closed sets.
But every Borel set is analytic and thus a con …
- 16.2k
4
votes
Accepted
Descriptive complexity of Hamel bases of R^ω
A projective Hamel basis under V=L should be easy: Take a $\Delta^1_2$ wellordering of $\mathbb R^\omega$ and prove the existence of a Hamel basis using this wellordering. That will give you a project …
- 16.2k
9
votes
Application of Fraïssé construction in set theory
While not strictly set theory, but rather a result connecting Ramsey theory and topological dynamics, two areas that are rather close to set theory, you might find the following example interesting:
…
- 16.2k
8
votes
Applications of infinite graph theory
Recently there has been quite a bit of activity in descriptive set theory concerning definable graphs.
Benjamin Miller derived several deep
classical results such as Silver's theorem (stating that eve …
- 16.2k