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Results tagged with descriptive-set-theory
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user 4600
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.
4
votes
Accepted
Measurability of $\{ x \in X ; H_0 x \subset A \}$
No, it is not true.
It suffices to show that the complement is not necessarily Borel.
Let $B=X\setminus A$, which is a general Borel set. We have
$$
\{x:H_0x\not\subseteq A\} = \{x:H_0x\cap B\ne\varn …
- 24.8k
9
votes
17
votes
2
answers
1k
views
Homeomorphisms and "mod finite"
Suppose $f:C\to C$ is a homeomorphism, where $C=\{0,1\}^{\mathbb N}$ is Cantor space.
Suppose $f$ preserves $=^*$ (equality on all but finitely many coordinates). Does it follow that $f$ also reflects …
- 24.8k
3
votes
Accepted
Generic sections of non-null sets are non-null
No, let $(r,y)\in B $ iff $r (0)=0$.
Then if $r $ is a random real with $r (0)=1$ you have a counterexample.
- 24.8k
3
votes
Accepted
Finding 1-generic paths through a tree $T \subseteq 2^{<\omega}$
What conditions can we impose on $T$ that guarantee $[T]$ contains a 1-generic member?
An element that is 1-generic relative to $T$ will not be on $[T]$ unless $[T]$ contains a whole clopen cone $[\ …
- 24.8k
8
votes
Examples of statements with a high quantifier complexity
In computability theory, a set of integers $A$ is infinitely often computably traceable if there is a computable function $h$ such that for all functions $g\le_T A$ ($g$ computable from $A$), there is …
- 24.8k
2
votes
Examples of statements with a high quantifier complexity
The Interchange lemma in the theory of formal languages has a $\Pi_5$ form:
$\forall L$, if $L$ is a context-free language, then
$\exists c$, $c$ is a positive integer, such that
$\forall n\ge 2$, $R …
- 24.8k
2
votes
Variously pointed closed sets
Let $\mu$ be a measure on $2^\omega$ which doesn't have a least Turing degree.
This exists by Theorem 4.2 of
Day, Adam R.; Miller, Joseph S., Randomness for non-computable measures, Trans. Am. Ma …
- 24.8k
2
votes
Can (how) one distinguish germs of continuous functions by a countable set of params?
The way I view it, germs of continuous functions are like
tails of infinite binary sequences, or
real numbers up to rational translation.
Finding "coordinates" on the space of germs would be like …
- 24.8k
8
votes
1
answer
252
views
Automorphisms of power set lattice mod finite
Let $N$ be a countably infinite set and let $\mathcal P$ denote power set.
I get that the automorphisms of $(\mathcal P(N),\subseteq)$ are all induced by permutations of $N$.
But what can be said abo …
- 24.8k
2
votes
Source on smooth equivalence relations under continuous reducibility?
Seems like this structure must be pretty complicated. For example, consider Brownian motion $\{W_t\}_{t\ge 0}$ with the equivalence relations
$$t\sim_\omega s\iff W_t(\omega)=W_s(\omega).$$
Here $\ome …
- 24.8k
4
votes
Can we define an "empirically generic" real number?
It sounds like you are talking about what in computability theory and set theory are known as Cohen generic reals (the lowest level of which in computability theory is 1-generic, then 2-generic and so …
- 24.8k