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Results tagged with descriptive-set-theory
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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.
7
votes
Accepted
Question about Woodin's paper "On the consistency strength of projective uniformization"
Rupert, the Martin-Solovay paper shows the absoluteness result follows form the existence of measurable cardinals. The measurables are used in the construction of certain trees (now called Martin-Solo …
- 32.5k
3
votes
The projection of a weakly homogeneous tree is determined
Not sure whether this is needed anymore, but:
The paper you want is John Steel's "The Derived Model Theorem." This paper gives a thorough and superb presentation of weak homogeneity and much more, in …
- 32.5k
15
votes
Definable Wellordering of the Reals
This is a very interesting topic! There are two ways to address the question. They are different, so I treat them separately.
I.
As mentioned by Stefan, knowing that we have a well-ordering of cert …
- 32.5k
13
votes
Accepted
Good source for Effective Descriptive Set Theory
A nice source is "Recursive aspects of descriptive set theory" by Mansfield and Galen Weitkamp, as mentioned by Yu in a comment. A problem with it is that it leaves out all the details of admissibilit …
- 32.5k
8
votes
Basis theorem (due to Solovay?)
The result is indeed Solovay's Basis Theorem.
It is a consequence of Moschovakis's Coding Lemma, and sometimes it is referred to as (a version of) the reflection theorem (for example, in section 8 o …
- 32.5k
8
votes
Accepted
$\Delta^1_2$-well ordering vs $\Delta^1_3$
Hi Yu,
No, your statement is equiconsistent with $\mathsf{ZFC}$. In
Leo Harrington. Long projective wellorderings, Annals of Mathematical Logic 12 (1977) 1-21, MR0465866 (57 #5752).
it is show …
- 32.5k
10
votes
Accepted
Failure of Shoenfield's Absoluteness
I'm turning my comment into an answer. With $\Sigma^1_2$ statements we can discuss well-foundedness: A real codes a well-founded model of enough set theory iff it codes a model (which is an arithmetic …
- 32.5k
5
votes
Exact consistency-strength of "all projective sets are Ramsey"
Hi David.
This is still open, and I don't know of any strategy that would result in a model with the property for all projective sets but not in a model with the property for all sets in $L({\mathbb …
- 32.5k
10
votes
Accepted
Does determinacy in $L(\mathbb{R})$ implies projective determinacy (in $V$)?
Suppose $M$ is an inner model (of $\mathsf{ZF}$) with the same reals as $V$, and let $A\subseteq \mathbb R$ be a set of reals in $M$. Suppose further that $A$ is determined in $M$. Under these assumpt …
- 32.5k
10
votes
Accepted
Sets of reals and absoluteness
At the projective level, there are nice level by level generalizations, and looking at Steel's paper in the Handbook should give you the proof and the pre-requisites to understand it fully. This is wh …
- 32.5k
13
votes
Accepted
Conflating reals and sets of countable ordinals "nicely"
The technique of almost disjoint forcing was introduced in
MR0289291 (44 #6482). Jensen, R. B.; Solovay, R. M. Some
applications of almost disjoint sets. In Mathematical Logic and
Foundations …
- 32.5k
4
votes
Accepted
Question about John Steel's "The derived model theorem"
Rupert, I will explain the argument for $Y=\omega$ (this makes no difference, but you may find it easier to visualize) when the continuous function is particularly nice (in a way I will make precise. …
- 32.5k
4
votes
Accepted
Proof of a soft version of Moschovakis's lemma
The argument you are looking for is given in Kanamori's book, see Theorem 28.15.
For the more nuanced version of the lemma, see section 7D in Moschovakis's descriptive set theory book (particularly …
- 32.5k
19
votes
Accepted
Perfect set property for projective hierarchy
Analytic sets have the perfect set property, provable in, say, ZF+DC. This goes back to Suslin, and is discussed in Kanamori's book "The higher infinite" (Around section 12).
Large cardinals imply tha …
- 32.5k
6
votes
Accepted
Models of Determinacy
Certainly, $L(\mathbb R)$ is not a mouse (rather, a weasel) over a countable set, and the only way I see of thinking of it as a mouse and still capturing all the reals is making it a mouse over $\math …
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