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Results tagged with descriptive-set-theory
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user 8133
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
Countable chain condition in $\text{BP}(X)$
Suppose I have uncountably many non-meager sets $\mathcal{A}=\{A_\eta\}_{\eta\in\omega_1}$ with the Baire property. Let $\mathcal{U}$ be a countable base for $X$.
Let $\mathcal{U}$ be a countable bas …
- 20.5k
4
votes
Accepted
Moschovakis Coding Lemma
I might be missing something, but I believe the answer is the following. First suppose $\Gamma$ isn't present. Then this just amounts to showing that, if $R\subseteq X\times Y$, $Z$ is a cofinite subs …
- 20.5k
4
votes
Games that never begin
A slightly more tangential answer, but one which I hope is still useful: there is a well-known connection between infinite games and infintary logic. In the usual context of games with no ending, dete …
- 20.5k
1
vote
Non-analytically measurable set in $\Delta^1_2$
You write:
I can prove that such a set exists but just wondering if there's a (fairly) concrete example.
The following might just be the proof you allude to, but I think it is fairly concrete:
A set …
- 20.5k
5
votes
Accepted
Descriptive set theoretic complexity of computable maps with respect to the Turing jump of t...
Every such set is both $F_\sigma$ and $G_\delta$, i.e. ${\bf \Delta^0_2}$.
The clear upper bound is that $Y$ is in $Z$ iff for some finite string $\sigma$ such that $\{a\}^\sigma(0)\downarrow=0$ we ha …
- 20.5k
5
votes
Is it inconsistent for a model of set theory to contain its own first order theory?
Adding to the existing answers, we can even have $A$ be $OD$ without resulting in contradiction. For example, for $\alpha$ a limit ordinal look at the real $$C_\alpha=\{i: 2^{\aleph_{\alpha +2i}}\not= …
- 20.5k
7
votes
Examples of independent $\Sigma_4^1$ statements
As a starting point, think about the sentence "There is a nonconstructible real." This is $\Sigma^1_3$ and clearly not downwards-absolute. However, it is upwards-absolute.
To get the desired situatio …
- 20.5k
7
votes
Accepted
Can it be that universal measurability is preserved by projections?
I'm not an expert, so please correct me if I'm wrong:
We can indeed continue the projective hierarchy beyond its finite levels. And like the Borel hierarchy, we can do this "from below" as follows: …
- 20.5k
8
votes
Accepted
A submodel of set theory with all reals which every set is analytic
In fact, the principle "Every set is analytic" is not consistent with $\mathsf{ZF}$ in the first place. We don't need choice to get a surjection $h$ from Baire space to the set of continuous maps on B …
- 20.5k
3
votes
Accepted
Intuition behind the non-Borel Lusin example
I definitely don't think this is a good "first example" of an analytic non-Borel set. It does, however, make more sense when put in comparison with the standard example of a co-analytic, non-Borel set …
- 20.5k
2
votes
Accepted
How much can complexities of bases of a "simple" space vary?
Welp, this wasn't my finest moment: unless I'm missing something, we can translate everything much more easily than I thought at first. (I may indeed be missing something, however, and I'll wait a whi …
- 20.5k
0
votes
Accepted
Spreading sets - especially without choice
I still don't know whether this has been studied before (and so references are still welcome!), but unsurprisingly, $Spread(\{$dominating families$\})=1$.
We begin with an easy lemma. (Note that in a …
- 20.5k
3
votes
Accepted
Comparing the sizes of uncountable sets of reals under AD
The following is due to John Steel - who is not on mathoverflow - following a suggestion (see Ordinal-definable witnesses to the perfect set property?) of Vladimir Kanovei; since none of this is my wo …
6
votes
Accepted
reverse mathematics of the Lebesgue measurability of analytic sets
I believe $\Delta^1_2$-CA$_0$ does indeed suffice; however, I don't see a way to pull this down to $\Pi^1_1$-CA$_0$.
This is contrary to my previous claim; my error was with respect to the strength o …
- 20.5k
0
votes
When does an "$\mathbb{R}$-generated" space have a short description?
This should be a comment, but it's too long - here's a $\mathsf{ZFC}+\mathsf{CH}$ example:
Let $\mathfrak{F}$ be the set of full-measure subsets of $\mathbb{R}$ (we could also take the set of non-mea …
- 20.5k