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Results tagged with descriptive-set-theory
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user 31807
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.
10
votes
Relation between projective hierarchy and universally measurable sets
Thanks, Joel, for mentioning our paper.
By Martin-Steel, if there exist infinitely many Woodin cardinals, then every uncountable projective set contains a perfect set, so, by the result of Hausdorff …
- 2,520
3
votes
Accepted
Scott Rank of Models of Infinitary Sentences
If $\phi$ is a counterexample to Vaught's Conjecture, then the Scott ranks of the models of $\phi$ include every limit ordinal below $\omega_{2}$ and above the quantifier depth of $\phi$. This follows …
- 2,520
6
votes
Accepted
Sets that are not $\infty$-Borel
Henle, Mathias and Woodin showed that if every set of reals is Ramsey, then forcing with $\mathcal{P}(\omega)/\mathrm{Fin}$ adds no new sets of ordinals. Any new set of reals in the extension (e.g., t …
- 2,520
4
votes
Accepted
Do escaping sets "uniformly" cover dominating sets under determinacy?
Here's something that seems to work. I can add more details if needed.
Suppose that every set of reals has the property of Baire. Then every function from $\omega^{\omega} \to \omega^{\omega}$ is cont …
- 2,520
3
votes
Accepted
A rather non-$F_\sigma$ Borel set
Here's an argument that the statement is false if the Continuum Hypothesis fails and the covering number for the null ideal is the same as the continuum. Wellorder the Borel sets of reals as $\langle …
- 2,520
13
votes
Accepted
Can the Turing degrees be linearly ordered?
You can't linearly order the Vitali ($\mathcal{P}(\omega)/\mathrm{Fin}$) degrees if every set of reals has the property of Baire, since you can't even choose between complementary degrees. The set of …
- 2,520
5
votes
Accepted
Comparing generic versions of $\mathbb{R}$
The answer seems to be no. Moreover: Suppose that every set of reals has the property of Baire. Let $\mathbb{C}$ be Cohen forcing and let $P$ be any wellorderable partial order. If $(c,d)$ is generic …
- 2,520