23
votes
Does ZF+AD settle the original Suslin hypothesis?
No. Because of silly reasons.
Recall that the powers of $\sf AD$ are quite limited to the world below $\Theta$. In particular, the proof that $\sf AD$ does not imply countable choice goes through ...
- 39.7k
19
votes
Accepted
Is determinacy on an infinite Dedekind finite set consistent?
The answer is no, you cannot have determinacy for all games on an infinite Dedekind-finite set. Indeed, one cannot even have clopen determinacy for games on such a set. So the answer to question 2 is ...
- 236k
18
votes
Accepted
Is there a minimal inner model for determinacy?
Assuming AD, Woodin showed that no inner model that is missing a real correctly computes $\omega_1$. (This almost follows from Theorem 9 of Velickovic-Woodin's "Complexity of the set of reals of ...
- 8,099
15
votes
Accepted
How much Dependent Choice is provable in $Z_2$? And what about Projective Determinacy?
The answer to the third question is yes. This is due to Kechris, who showed that $\text{ACA}_0$ plus schematic PD implies schematic DC. This appears in the last section of "The Axiom of ...
- 8,099
13
votes
Accepted
The Axiom of Determinacy and the Banach-Mazur game
The claim is false. The Banach-Mazur game, also known as the $**$-game shows (and is equivalent to) that every set of reals has the Baire property. What is true, as you've noted, is that if one has a ...
- 3,804
13
votes
Accepted
Can the Turing degrees be linearly ordered?
You can't linearly order the Vitali ($\mathcal{P}(\omega)/\mathrm{Fin}$) degrees if every set of reals has the property of Baire, since you can't even choose between complementary degrees. The set of $...
- 2,520
13
votes
How much Dependent Choice is provable in $Z_2$? And what about Projective Determinacy?
Here is a natural model $M$ of $Z_2$ where projective DC fails. Starting with a model of ZFC + $V=L$, force over $L$ with the Levy collapse $\mathrm{Coll}(\omega,{<\aleph_\omega})$ collapsing all ...
- 9,902
12
votes
Accepted
How much determinacy do you need for second order arithmetic to be as strong as ZFC?
Because ZFC proves soundness of $\text{Z}_2$, no consistent finite extension of $\text{Z}_2$ proves all second order arithmetic statements that are provable in ZFC (for example, the statement "...
- 7,525
12
votes
Accepted
Determinacy and Woodin cardinals
The result is true if you actually collapse the Woodin cardinal, not just the cardinals smaller than it. This follows from the results in Itay's chapter in the Handbook. See MR2768701, zbM1198.03057
...
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11
votes
Cardinal arithmetic under determinacy
Starting from $L(\Bbb R)$, we can take a symmetric extension which preserves $\sf DC$ and adds an $\omega_1$-amorphous set, just somewhere far above $\Theta$.
So we cannot prove that (1) or (2) hold ...
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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 ...
- 32.5k
10
votes
Does ZF+AD settle the original Suslin hypothesis?
Nice question! But which aged set theorist drinks prominent Scotch?
The following does not answer this question either. Let $T \in L$ be your favorite Suslin tree of $L$. Consider the $L({\mathbb R})$...
- 1,808
10
votes
Determinacy and Woodin cardinals
(This answers what was the original question, which was with $\mathrm{Coll}(\omega,{<\kappa})$ replacing $\mathrm{Coll}(\omega,\kappa)$.)
Hmm...doesn't this contradict Theorem 1.22 of "MICE ...
- 9,902
9
votes
Accepted
Existence of a winning strategy in the $\boldsymbol{\Delta}_2^0$ game
Player I has a winning strategy: First play a singleton $A_0=A_1=\ldots=\{z_0\}$, for some real $z_0$, and the $x_n$'s consistent with $z_0$, until player II plays their first 1, if they ever do. ...
- 9,902
9
votes
What is the complexity of the winning condition in infinite Hex? In particular, is infinite Hex a Borel game?
The complexity is arithmetic, and at most $\Sigma^0_7$.
The proof is based on showing that my counterexample to PyRulez's proposed formula is essentially the only one.
Let $G$ be the set of tiles of ...
- 740
9
votes
Large cardinals in ZF + DC + AD
An unpublished theorem due to Woodin (which appears without proof as Theorem 7.35 of "In search of Ultimate L") states that if the $\Omega$ conjecture holds and there is a proper class of ...
- 8,099
8
votes
Accepted
Strong limit cardinals in AD
First, let me point out that the notion of "high limit cardinal" is misleading, and perhaps not quite what you want it to be.
The reason is that strong limit cardinals are defined, in the absence of ...
- 39.7k
8
votes
Does ZF+AD settle the original Suslin hypothesis?
Concerning the existence of Suslin trees under $\mathsf{AD + V = L(\mathbb{R})}$:
Assume $ZF + AD^+ + V = L(\mathscr{P}(\mathbb{R}))$. Let $(T,\prec)$ be an $\omega_1$-tree with all levels countable. ...
- 1,750
8
votes
Accepted
Getting measures (especially on $\omega_2$) from potential clubs
The claim you are trying to prove is false: $L(\mathbb R)$ has no measurable cardinals greater than $\Theta$. Work in $L(\mathbb R).$ We will use Woodin's theorem that $\text{HOD} = L[\mathbb P]$ for $...
- 8,099
8
votes
Accepted
Aronszajn Trees when AC fails
Is it acceptable? Sure. In some sense, it is an Aronszajn tree.
The condition of being well-founded, which in the presence of $\sf DC$ is the same as saying there are no decreasing sequences, is ...
- 39.7k
8
votes
Accepted
Cardinal arithmetic under determinacy
Simon Thomas "Superrigidity and countable Borel equivalence relations" Corollary 4.9 gives a countable Borel equivalence relation E shows E and 2E are not Borel bireducible. (The examples ...
- 96
7
votes
Accepted
Uniformization under AD
Here's an argument under the further assumption that $V=L(\mathbb{R})$. Something like this is presumably recorded somewhere. The main point comes from Steel's paper "Scales in $L(\mathbb{R})$&...
- 9,902
7
votes
Accepted
What sets can be unraveled?
I emailed Itay Neeman, and he told me the following:
As far as I know it's open. I don't think anything is known about
unraveling beyond what you can get from my methods. These give the
Suslin ...
7
votes
Accepted
Strong partition property + DC + existence of non-principal ultrafilter on $\omega$
The key reference for this is
MR0799042 (87d:03141). Henle, J. M.; Mathias, A. R. D.; Woodin, W. Hugh. A barren extension. In Methods in mathematical logic (Caracas, 1983), C. A. Di Prisco, editor, ...
- 32.5k
7
votes
Accepted
Is $Z_3+{\rm AD}$ equiconsistent with $\text{ZFC+AD}^{L(\mathbb{R})}$
The only reference I know for precisely these matters is the handbook chapter
MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, ...
- 32.5k
7
votes
Accepted
Limitations of determinacy hypotheses in ZFC
Here is a non-exhaustive list of limitations. I'm including some from ZF alone, for completeness.
As you mentioned, naturally there is a non-determined game of length $\omega$, in ZFC.
Without ...
- 246
7
votes
Accepted
Is there a natural inner model of AD$_\mathbb{R}$?
A Wadge initial segment (of $\mathcal P(\mathbb R)$) is a subset $\Gamma$ of $\mathcal P(\mathbb R)$ such that whenever $A\in\Gamma$ and $B\le_W A$, where $\le_W$ denotes Wadge reducibility, then $B\...
- 32.5k
7
votes
Which forcings preserve (some) determinacy?
Bumping to mention/advertise a recent development:
Today, William Chan and Stephen Jackson posted this paper to the arxiv. They proved that a broad class of forcings (under reasonable hypotheses) ...
- 21.1k
6
votes
How to prove projective determinacy (PD) from I0?
Chapter 6 of the paper Large Cardinals and Projective Determinacy
gives a proof of $PD$ from $I_0$.
- 32.1k
6
votes
Accepted
Under ZF + DC + AD, is it known what the properties are of the Hartogs number for $\mathcal{P}(\kappa)$ for some $\kappa>\aleph_0$?
This is a bit confused, but I think what you're asking is:
Let $S(\alpha)$ be the supremum of all the ordinals onto which $\mathcal{P}(\alpha)$ surjects. What can we say about e.g. $S(\omega_1)$?
(...
- 21.1k
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