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 ...
Asaf Karagila's user avatar
  • 37.8k
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 ...
Joel David Hamkins's user avatar
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 ...
Gabe Goldberg's user avatar
15 votes

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 ...
Gabe Goldberg's user avatar
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 ...
Rachid Atmai's user avatar
  • 3,746
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 $...
Paul Larson's user avatar
  • 2,560
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 ...
Farmer S's user avatar
  • 8,627
12 votes
Accepted

Ordinal-definable witnesses to the perfect set property?

Let $A$ be the set of those reals that are not OD. Then $A$ is OD (since I've just defined it), and it's uncountable (since its complement, being a well-orderable set of reals, must be countable ...
Andreas Blass's user avatar
11 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 "...
Dmytro Taranovsky's user avatar
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 ...
Andrés E. Caicedo's user avatar
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 $L({\mathbb R})$...
Ralf Schindler's user avatar
9 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. ...
William's user avatar
  • 1,830
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. ...
Farmer S's user avatar
  • 8,627
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 ...
Ilkka Törmä's user avatar
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 ...
Gabe Goldberg's user avatar
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 ...
Asaf Karagila's user avatar
  • 37.8k
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 $...
Gabe Goldberg's user avatar
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 ...
Asaf Karagila's user avatar
  • 37.8k
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 $L(\mathbb{R})$&...
Farmer S's user avatar
  • 8,627
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, ...
Andrés E. Caicedo's user avatar
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, ...
Andrés E. Caicedo's user avatar
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) ...
Noah Schweber's user avatar
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 ...
Juan's user avatar
  • 246
6 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\...
Andrés E. Caicedo's user avatar
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 $I_0$.
Mohammad Golshani's user avatar
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. $S(\omega_1)$? (...
Noah Schweber's user avatar
6 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 ...
6 votes
Accepted

How much of second-order arithmetic do you need for $\mathbf{\Sigma}^1_1$-determinacy to give you countable transitive models of $\mathsf{ZFC}$?

$\mathsf{RCA}_0 + \mathbf{\Sigma}^1_1\text{-Det}$ suffices to get sharps for all reals (and thus ctm of ZFC and more). With boldface determinacy principles, we can bootstrap the background theory. $\...
Dmytro Taranovsky's user avatar
5 votes
Accepted

Comparing generic versions of $\mathbb{R}$

The answer seems to be no. Moreover: Suppose that every set of reals has the property of Baire. Let $\mathbb{C}$ be Cohen forcing and let $P$ be any wellorderable partial order. If $(c,d)$ is generic ...
Paul Larson's user avatar
  • 2,560
5 votes
Accepted

Pointwise definable models of determinacy

The least ordinal $\kappa$ such that $L_\kappa(\mathbb R)$ satisfies KP is a counterexample. This essentially reduces to Theorem 1.3 of Tony Martin's "The Largest Countable This, That, and the Other." ...
Gabe Goldberg's user avatar

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