All Questions
Tagged with determinacy set-theory
86 questions
3
votes
1
answer
280
views
Does determinacy in $L(\mathbb{R})$ implies projective determinacy (in $V$)?
Does $AD^{L(\mathbb{R})}$ directly implies projective determinacy? At least it certainely implies $PD$'s consistency.
- 615
26
votes
3
answers
2k
views
Does ZF+AD settle the original Suslin hypothesis?
Everyone knows that the real line $\langle\mathbb{R},<\rangle$ is
the unique endless complete dense linear order with a countable
dense set. Suslin's
hypothesis is
the question whether we can ...
- 236k
7
votes
1
answer
555
views
Limitations of determinacy hypotheses in ZFC
When considering (set-theoretic) games, we have three parameters we can adjust:
Definability of the payoff set
The set of legal moves
The length of the game
When working in $\textsf{ZFC}$, what are ...
- 1,170
12
votes
1
answer
477
views
Is there a natural inner model of AD$_\mathbb{R}$?
The question is as in the title, but let me explain a bit.
Assuming a proper class of Woodin cardinals, $L(\mathbb{R})$ satisfies AD (and DC). And $L(\mathbb{R})$ is a very natural inner model. I'm ...
- 21.1k
11
votes
1
answer
581
views
Is determinacy on an infinite Dedekind finite set consistent?
Consider $\mathrm{AD}_X$, determinacy for games where players pick moves from $X$. We know that it is consistent for $X = \omega$ or $\mathbb{R}$ (under large cardinal assumptions), but inconsistent ...
- 643
6
votes
1
answer
402
views
How much real determinacy can live in $L(\mathbb{R})$?
It's well-known that AD$_\mathbb{R}$ fails in $L(\mathbb{R})$, provably in ZFC. This is because:
AD$^{L(\mathbb{R})}$ implies DC$^{L(\mathbb{R})}$.
Over ZF+DC, AD + "Every set of reals has a scale" ...
- 21.1k
5
votes
0
answers
192
views
The club filter in definable preorders
So this is an embarrassing question. Call a preorder $\mathbb{P}$ good if it has the following properties:
Every countable chain in $\mathbb{P}$ has a least upper bound.
$\mathbb{P}$ is directed (any ...
- 21.1k
1
vote
1
answer
366
views
Defining cones and Turing cones
In Set Theory Jech defines a cone to be a subset of the Baire Space $\mathcal{N}$ of the form
$$\operatorname{cone}(x_0)= \{x : x_0 \in L[x]\}$$
where $x_0 \in \mathcal{N}$. Jech then defines the ...
- 180
6
votes
1
answer
400
views
$\operatorname{AD}$ and the measurability of $\omega_1$
Are there proofs of the measurability of $\omega_1$ (under $\operatorname{AD}$) that do not use Turing degrees nor the $\Sigma_1^1$ boundedness lemma?
I've been struggling to find an "elementary" ...
- 180
5
votes
0
answers
395
views
Why is this transfinite game not determined?
This question originates from the paper On the Axiom of Determinateness by Jan Mycielski, section 7. Given a set $X$ and an ordinal $\alpha$, the author defines a transfinite game of length $\alpha$ ...
- 907
5
votes
1
answer
231
views
Spreading sets - especially without choice
For what follows, I work in ZF+AD+DC. However, the questions below are not obviously trivial in ZFC, so I'm also interested in results in that system.
Suppose I have a set $X\subseteq \mathbb{R}$. ...
- 21.1k
5
votes
1
answer
471
views
Comparing the sizes of uncountable sets of reals under AD
Working in ZF+AD, let $$\theta_0(X)=\min\{\alpha\in ON: \not\exists f: X\rightarrow \alpha\mbox{ surjective and OD}\}$$ be the least ordinal onto which $X$ does not surject in an OD way, for $X\...
- 21.1k
6
votes
1
answer
302
views
Ordinal-definable witnesses to the perfect set property?
This possibly a very basic descriptive set-theory question; if it is too basic for MO, feel free to migrate.
Throughout we work in ZF+AD. My question is:
If $A$ is an uncountable OD set of reals, ...
- 21.1k
10
votes
0
answers
306
views
The Chang model after collapsing an inaccessible limit of Woodins
If $\kappa$ is an inaccessible cardinal and $G \subset \operatorname{Col}(\omega,\mathord{<}\kappa)$ is a $V$-generic filter, then in $V[G]$ the Chang model $L(\text{Ord}^\omega)$ satisfies "every ...
- 5,471
10
votes
1
answer
419
views
Assuming AD, is every infinite cardinal closed under power set in a choice model?
Assume AD+DC. Assume $\kappa$ is an infinite cardinal and $N$ is a (set or class) transitive model of ZFC containing $\kappa$.
Is it true that for all $\alpha<\kappa$, $N$ thinks that the power ...
- 685
7
votes
0
answers
239
views
Countable choice in $L(\mathbb{R}^*_G)$
Let $\lambda$ be a singular strong limit cardinal and let $G \subset \text{Col}(\omega,\mathord{<}\lambda)$ be a $V$-generic filter. Let $\mathbb{R}^*_G = \bigcup_{\alpha < \lambda} \mathbb{R}^{...
- 5,471
9
votes
0
answers
271
views
Which forcing types preserve the axiom of determinacy?
Do we have some rudimentary understanding of some properties that a forcing can have in order to guarantee that it doesn't violate the axiom of determinacy?
To be more specific, in Which forcings ...
- 39.7k
5
votes
1
answer
500
views
What axioms (other than choice) have a taming effect on the ordering of cardinalities?
Axiom of choice arranges all cardinalities into a well-ordered chain but without it their ordering can be wild in general ZF models, e.g. two cardinalities may not even have inf or sup. However, ...
- 1,731
5
votes
1
answer
651
views
$\omega$ universally Baire sets, tree representations
I've recently encountered the notion of a universally Baire set, and I've tried to look at the paper by Feng, Magidor and Woodin where this notion is studied. There are several points that confuse me.
...
- 51
5
votes
1
answer
258
views
Forcing over models of determinacy
Consider a ctm $\mathfrak{M}$ of $ZF+AD^+$. Is it possible to force over $\mathfrak{M}$ to get a model of ZFC which satisfies further the following:
Every projectively definable family of sets of ...
- 1,416
9
votes
1
answer
229
views
n odd: $\bf\Delta^1_n$ wadge degrees are $< \bf\delta^1_{n+1}$
My adviser is out of town and there is a comment in the Van Wesep paper "wadge degrees and descriptive set theory" that I can't figure out.
Work in ZF+AD throughout.
As stated in the title, the ...
- 603
9
votes
1
answer
306
views
$\Sigma^0_1\wedge\Pi^0_1$-Determinacy holds in the second admissible above the game
Let $T$ be a game tree and $T\in N\in M$, where $N,M$ are the two least admissibles containing $T$. Let $A$ be a boolean combination of two lightface open sets in $[T]$, or alternatively, a boolean ...
- 437
8
votes
1
answer
412
views
Universal $(\Sigma^2_1)^{\text{Hom}_{\mathord{<}\lambda}}$ set
Does anyone know of a reference for the fact that if $\lambda$ is a limit of Woodin cardinals, then the pointclass $(\Sigma^2_1)^{\text{Hom}_{\mathord{<}\lambda}}$ is $\omega$-parameterized? By ...
- 5,471
7
votes
1
answer
588
views
Determinacy from $\omega_1\rightarrow(\omega_1)^{\omega_1}$
Assuming the Axiom of Determinacy (abbreviated AD), Martin showed how to derive a rather strong partition on $\omega_1$, namely that $\omega_1\rightarrow(\omega_1)^{\omega_1}$. In "Infinitary ...
- 333
13
votes
1
answer
791
views
Consistency strength of projective determinacy (PD)
Let PD stand for projective determinacy, and consider the two claims:
(1) For each n=1,2,..., Con(ZFC+PD) implies Con(ZFC + there are n Woodin cardinals)
(2) Con(ZFC+PD) implies Con(ZFC + there are ...
- 205
12
votes
1
answer
805
views
Coding a model of $0^\sharp$ from a $\Pi^1_1$ Gale-Stewart game
As a preface to this question, this is my first time asking on Math overflow, and this seemed like the sort of question that would be acceptable here. However, I apologize if it is not.
A method for ...
- 333
35
votes
3
answers
5k
views
Counterintuitive consequences of the Axiom of Determinacy?
I just read Dr Strangechoice's explanation that if all subsets of the real numbers are Lebesgue measurable, then you can partition $2^\omega$ into more than $2^\omega$ many pairwise disjoint nonempty ...
- 7,633
11
votes
2
answers
708
views
Which forcings preserve (some) determinacy?
The question is exactly as in the title. I'm interested in general in all questions of the form "which forcings preserve property P?" for any P, but determinacy assumptions occupy a special place in ...
- 21.1k
6
votes
1
answer
531
views
sigma-algebra generated by OD sets
Assume $V=L(\mathbb{R})$ and the Axiom of Determinacy. Is every set of reals generated by ordinal-definable sets of reals under the operations of countable union and intersection?
The class of sets ...
- 5,471
7
votes
2
answers
547
views
Weakly homogeneous trees under AD
If AD$_\mathbb{R}$ holds and $\kappa < \Theta$ then every tree $T$ on $\kappa$ is weakly homogeneous (Martin–Woodin, "Weakly homogeneous trees.") I recall hearing that the hypothesis can be ...
- 5,471
16
votes
1
answer
697
views
How to prove projective determinacy (PD) from I0?
Martin and Steel (in 1987?) showed that if there are infinite many Woodin cardinals then every projective set of reals is determined (PD).
However, it is mentioned in many texts that in 1983/1984 ...
- 779
9
votes
1
answer
677
views
Consistency strengths related to the perfect set property
I want a model of $\mathrm{MA}_{\sigma\mathrm{-centered}}+\neg\mathrm{CH}$ in which every set of reals in $L(\mathbb{R})$ has the perfect set property. In terms of consistency strength, it is known ...
- 1,101
5
votes
1
answer
677
views
value of Theta in ZF+AD
Since I found out about it, I've always been interested in the Axiom of Determinacy rather than the Axiom of Choice. Along these lines, I've kept flipping back to http://en.wikipedia.org/wiki/%CE%98_%...
user5810
9
votes
2
answers
1k
views
Martin's cone theorem and recursion theory
Martin's remarkable cone theorem in the theory of determinacy says the following:
Suppose $A\subseteq \omega^\omega$ is Turing invariant and determined. If $\forall x\exists y(x\le_T y\& y\in ...
- 32.5k
10
votes
5
answers
1k
views
Measurable cardinals under Axiom of Determinacy
I seem to remember reading somewhere that ZF+AD proves that $\omega_1$ and $\omega_2$ are measurable cardinals.
Is that right?
If so, can someone [point me to or give here] a [sketch or proof] of ...
user5810
3
votes
3
answers
696
views
Determinacy interchanging the roles of both players
Let me refer to Jech's "Set Theory" Chap. 33 Determinacy:
"With each subset A of $\omega^\omega$ we associate the following game $G_A$, played by two players I and II. First I chooses a natural ...
- 1,465