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Results tagged with descriptive-set-theory
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user 6794
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
Accepted
Illfounded trees as "retract" of all trees
Suppose, toward a contradiction, that $R$ is a function of the sort you asked about. Then, for any tree $T$, we have the equivalence "$T$ has an infinite path $\iff$ every infinite path through $R(T) …
- 73.2k
5
votes
Code universal arithmetical sets by a hyperarithmetical set?
Perhaps the quickest answer to the original question is that the clause "If we define the sets $S_n$ with care" makes things look worse than they are. If you use any of the standard constructions of …
- 73.2k
13
votes
Continuously selecting elements from unordered pairs
This should have been a comment, but it got a bit too long.
A possibly useful necessary condition, in regular spaces, for the existence of a continuous selector is that there should not exist three …
- 73.2k
5
votes
Models of $AD$ different from $L(\mathbb{R})$
I'm not sure what you mean by "genuine models", but let me comment on how different $L(\mathbb R)$ is from $V$. They look very different to me. Partly this is because the axiom of choice holds in $V …
- 73.2k
17
votes
Accepted
Does there exist an uncountable separable metric space $X$ such that every subset of $X$ is ...
Under Martin's Axiom plus the negation of CH, every set $X$ of reals of size $<\mathfrak c$ is a Q-set, which means that every subset of $X$ is an $F_\sigma$-set with respect to the subspace topology …
- 73.2k
5
votes
Accepted
When can the one-one continuous image of a perfect set fail to be perfect?
A perfect subset of a space $X$ is required not only to have no isolated points but also to be closed in $X$. Compactness of $P$ is used to ensure that $f[P]$ is (compact and therefore) closed.
- 73.2k
12
votes
Accepted
What is the descriptive complexity of a set added by Cohen forcing?
Since Cohen forcing is weakly homogeneous, all hereditarily ordinal definable sets in the Cohen extension are already in the ground model. That applies in particular to any ordinal definable real, an …
- 73.2k
4
votes
Are Vitali-type nonmeasurable sets determinate?
To answer the last (parenthetical) piece of the question, which Joel seems to have ignored: Density doesn't help; a dense Vitali set $V$ can still have an easy winning strategy for Bob. The point is …
- 73.2k
9
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Follow up question: Shelah's "Can you take Solovay's inaccessible away?"
Gabe Goldberg has already given a couple of proofs, but here's another one, just because I like it. Being a model of ZFC, $L[x]$ sees a one-to-one map of its $\omega_1$ into its reals. That map, in t …
- 73.2k
23
votes
Accepted
Partitioning $\mathbb{R}$ into $\aleph_1$ Borel sets
It suffices to express $\mathbb R$ as the union of $\aleph_1$ (not necessarily disjoint) Borel sets such that no countably many of them cover $\mathbb R$, because then you can list them in an $\omega_ …
- 73.2k
14
votes
Accepted
Do Borel subsets of the plane with null sections have Borel projections?
Notice that $\omega^\omega$ can be embedded to a null subset of itself by sending any sequence $(a_0,a_1,a_2,\dots)$ to $(a_0,0,a_1,0,a_2,0,\dots)$. So any Borel phenomenon that can happen in $\omega^ …
- 73.2k
4
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Limits of determinacy on reals
The statement that every subset of $\mathbb R^\omega$ is determined is called $AD_{\mathbb R}$, and it's consistent relative to large cardinals. I don't remember exactly how large, but I vaguely reca …
- 73.2k
9
votes
Accepted
Measurably-isomorphic subsets of polish spaces and the continuum hypothesis
Every Borel subset (and in fact every analytic subset) of a Polish space either is countable or has a perfect subset. In particular, an uncountable Borel subset in a Polish space has the cardinality …
- 73.2k
7
votes
Accepted
Subset of the reals with zero inner measure and "full" outer measure in $\mathsf{ZF}+\mathsf...
I'll start as in Monroe Eskew's answer: Assume $X$ isn't measurable and get Borel sets $A,B$, with $A\subseteq X\subseteq B$, such that the measure of $A$ (resp. $B$) is the inner (resp. outer) measur …
- 73.2k
7
votes
Accepted
Iteration of random reals
Since $x$ is random over $V$, the fact that $C_x^{V[x]}$ is null is (like any fact about $x$ in $V[x]$) forced by some condition in $V$. This condition is the equivalence class, modulo the null ideal …
- 73.2k