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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.
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Moschovakis Coding Lemma
I trying to study the Coding Lemma (in descriptive Set Theory) and there is a small point in the proof that I don't understand. Let me first recall the version I'm studying ( there are different versi …
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Theorems proved with AD whose proof is also known in the ZF world
This question arises from discussions with my professor and from Todd Eisworth comments in this question Large cardinal axioms and the perfect set property
In $L(\mathbb{R})$ we have $AD$ and it is a …
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Borel reduction/Wadge hierarchy
An equivalence relation $E$ is Borel-reducible to an equivalence relation $F$ if there is a $\Delta^1_1$ function $f$ such that $xEy$ holds iff $f(x)Ff(y)$ holds. A set $A\subset \omega^{\omega}$ is W …
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Models of $AD$ different from $L(\mathbb{R})$
Today it is known that $AD$ (the axiom of determinacy of games played with integers) is true in $L(\mathbb{R})$. Has it been proven that this is the only model in which $AD$ is true? Have other models …
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Kunen tree and Martin tree
Do we know under which conditions the Kunen tree (Recall the Kunen tree provides an analysis of the equivalence classes of functions $f: \omega_1 \to \omega_1$ with respect to the normal measure $W^1_ …
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Large cardinal axioms and the perfect set property
It is known that if there is a measurable cardinal then every $\Pi_1^1$ set has the perfect set property (i.e it is either countable or contains a copy of $2^{\omega}$). Also if we have $\Pi_1^1$-dete …
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Sets of reals and absoluteness
Schoenfield's absoluteness states that if $\phi$ is $\Sigma^1_2$ then $V\models \phi$ iff $L\models \phi$. The set of reals in $L$ is $\Sigma^1_2$ and it is the largest countable $\Sigma^1_2$ set of r …
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Why does inner model theory need so much descriptive set theory (and vice versa)?
I am curious about how much descriptive set theory is involved in inner model theory.
For instance Shoenfield's absoluteness result is based on the construction of the Shoenfield tree which projectio …
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A result of Steel on characterizing lightface pointclasses
In the article Projectively wellordered inner models, Steel proves the following theorem (4.12):
Theorem: Let $n < \omega$ and suppose $\mathcal{M}_n^{\sharp}$ exists. Let $\delta=\delta^1_{n+2}$. T …
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Models of Determinacy
Today we have that $L(\mathbb{R}) \models AD$ (assuming there are $\omega$ many Woodin cardinals and a measurable above them all). I was wondering what other models of $AD$ might look like and if it i …
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