Questions tagged [genericity]
Questions about notions of genericity in computability theory/descriptive set theory. Not restricted to the 'standard' partial order producing $\alpha$-generics. Use the forcing tag for set-theoretic forcing.
7 questions
6 votes
3 answers
538 views
$\omega$-branching tree of mutual generics
Is there an $\omega$-branching tree $T \subset \omega^{< \omega}$ without terminal nodes such that any pair $f \neq g \in [T]$ are mutually generic? Let's say mutually $1$-generic but I'm ...
- 3,987
5 votes
1 answer
309 views
Splitting 0' into two 1-generic reals
I have come across multiple research papers where they have mentioned that two 1-generic reals $x$ and $y$ can be constructed such that $x \oplus y \equiv_{T} \textbf{0}'$, where $\textbf{0}'$ is the ...
user527095
1 vote
0 answers
58 views
Are the $\omega$-generic arithmetic degrees downward closed
A degree is $\alpha$-generic if it has representative that is $\alpha$-generic. Are the $\omega$-generic arithmetic degrees (i.e. the degree structure induced by arithmetic reproducibility) downward ...
- 3,987
3 votes
1 answer
276 views
What's the measure of all 1-generic sets?
A set $A$ is 1-generic if it forces its jump, namely for any $e\in\omega$, there exists $\sigma\preceq A$ such that: $\Phi^{\sigma}_{e}(e)\downarrow\vee(\forall\tau\succeq\sigma)(\Phi^{\tau}_{e}(e)\...
- 33
2 votes
1 answer
166 views
Cite for fact that every r.e. degree bounds a 1-generic
Odifreddi doesn't give a cite (at least in proposition XI.2.10) for the proposition that every non-zero r.e. degree computes a 1-generic. What paper should I cite for this proposition?
- 3,987
2 votes
0 answers
229 views
Genericity of an induced projection map
I am cross-posting a question asked on Math Stackexchange that has not been answered, in which I am still interested in. Let $X,Y$ be smooth manifolds, $S'$ a submanifold of $Y$, and $f:\mathbb{R}\...
- 31
0 votes
1 answer
126 views
Kurtz randomness and supermartingales with infinite *limit*
Suppose you replace the usual success conditions for a supermartingale (lim sup is infinite) with the requirement that the actual limit is infinite, e.g. a supermartingale $B$ succeeds on $X \in 2^\...
- 3,987