Questions tagged [determinacy]
The determinacy tag has no usage guidance.
96 questions
2
votes
0
answers
59
views
Consistency of Sigma-V-2 uniformization with AD
Is ZF + AD consistent with: For every real $r$, every true $Σ^V_2(r)$ statement has a $Δ^V_2(r)$ example?
DC is provable in ZF + every true $Σ^V_2$ statement has a $Δ^V_2$ example (i.e. witness). ...
- 7,545
4
votes
0
answers
107
views
Partial uniformization under AD
Under ZF + AD, and especially $\text{AD}^+$, even if uniformization fails for reals, in some ways it must almost hold.
For a notion of small, we say that uniformization holds on a co-small set of ...
- 7,545
7
votes
0
answers
259
views
A version of determinacy for all sets
Under ZF + AD, some games are undetermined because of lack of choice. In fact, the axiom of choice is equivalent to determinacy of games of length 2. However, we can ask whether the lack of choice ...
- 7,545
4
votes
0
answers
205
views
Fine structure without choice
In set theory, are there approaches to fine structure that give fine-structural models that do not satisfy the axiom of choice?
We can build fine-structural models above a given set (such as $\mathbb ...
- 12.9k
4
votes
1
answer
238
views
AD and simultaneous well-orderability principle
Is the axiom of determinacy (AD) consistent with the following choice principle, and if yes, does it hold in $L(ℝ)$ under AD:
Simultaneous well-orderability: For every function $f:P(Ord)→\text{...
- 56
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 ...
- 2,031
10
votes
2
answers
564
views
Cardinal arithmetic under determinacy
Work in a reasonable theory of determinacy such as $\mathsf{ZF+DC+AD}$. Which of the following identities are true for arbitrary infinite sets?
$|A^2|=|A^3|$ (motivated by an MSE question that asks ...
- 96
9
votes
2
answers
455
views
Determinacy and Woodin cardinals
I am looking for a reference for the following result:
Theorem Assume $\kappa$ is a Woodin cardinal. Then after forcing with the Levy collapse $\mathbb{P}=Col(\omega, \kappa)$, the $\Sigma^1_2$-...
- 9,902
3
votes
0
answers
212
views
Periodicity in the cumulative hierarchy
Under Reinhardt cardinals in ZF, the cumulative hierarchy exhibits a periodicity in that for large enough $λ$, certain properties of $V_λ$ depend on whether $λ$ is even vs odd. See Periodicity in the ...
- 7,545
2
votes
0
answers
118
views
Uniformization and functions on Turing degrees
Assuming Martin's Conjecture on functions between Turing degrees, is AD + DC consistent with existence of an $f:\mathcal{D}_t → \mathcal{D}_t$ of rank $Θ$ ?
$\mathcal{D}_t$ is the set of Turing ...
- 7,545
6
votes
0
answers
125
views
From HODs to corresponding models of AD
If $M$ is HOD of a model $N$ of $\text{AD}^+ + V=L(P(ℝ))$, what kind of forcing construction in $M$ gives back such an $N$?
HODs for $\text{AD}^+ + V=L(P(ℝ))$ are conjectured (and under anti-large ...
- 7,545
6
votes
0
answers
185
views
Complexity of transfinite 5-in-a-row and other games
Suppose that 5-in-a-row is played on an infinite board, and after an infinite number of moves, if no one won yet and there is an empty square, the game just continues. At limit steps, it is the first ...
- 7,545
4
votes
0
answers
116
views
Closure of a pointclass under universal real quantification
Let us assume $\mathsf{AD}^+$ and let $\Gamma$ be a pointclass such that $P(\mathbb{R})\cap L(\Gamma)=\Gamma$ and $L(\Gamma)\models\mathsf{AD}_\mathbb{R}+\mathsf{DC}$. Since the cofinality of $o(\...
- 41
7
votes
3
answers
460
views
How much Dependent Choice is provable in $Z_2$? And what about Projective Determinacy?
So, second order arithmetic, $Z_2$, is capable of proving quite a few things. One thing which would be of use is dependent choice for $\mathbb{R}$.
Basically, dependent choice on $\mathbb{R}$ says ...
- 17.7k
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 ...
- 4,713
7
votes
0
answers
198
views
Strengthening Determinacy in constructive set theory?
Recall how games work. Let $X$ be a set (the "game space") and $\alpha$ an ordinal (the "game clock"). Alice and Bob take turns naming elements of $X$. We write them down in order ...
- 63.9k
10
votes
1
answer
255
views
How much of second-order arithmetic do you need for $\mathbf{\Sigma}^1_1$-determinacy to give you countable transitive models of $\mathsf{ZFC}$?
This is in some sense a follow-up to this question.
The answer there says that over $\mathsf{Z}_2$ (second-order arithmetic), (boldface) $\mathbf{\Sigma}^1_1$-determinacy is enough to entail the ...
- 7,545
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
13
votes
1
answer
524
views
How much determinacy do you need for second order arithmetic to be as strong as ZFC?
From Wikipedia (I couldn't find the original source):
$\text{ZFC} + \{\text{there are $n$ Woodin cardinals: $n$ is a natural number}\}$ is conservative over $\text{Z}_2$ with projective determinacy.
...
- 7,545
24
votes
2
answers
1k
views
What is the complexity of the winning condition in infinite Hex? In particular, is infinite Hex a Borel game?
Consider the game of infinite Hex, where two players Red and Blue alternately place their stones on the infinite hex grid, each aiming to create a winning configuration.
Red wins after infinite play, ...
- 740
2
votes
0
answers
267
views
Is determinacy of (some) very long open games consistent?
For $\varphi$ a first-order sentence in the language of set theory and $\kappa$ an ordinal, let $G_\varphi^{\kappa}$ be the game of length $\kappa$ in which players $1$ and $2$ alternately play ...
- 21.1k
7
votes
0
answers
258
views
Is this determinacy principle consistent?
Let $\mathsf{ODet}_{\omega_1}(L(\mathbb{R}))$ be the following principle ("determinacy for simple open length-$\omega_1$ games"):
If $\kappa$ is any ordinal and $X\subseteq \kappa^{<\...
- 4,713
5
votes
1
answer
1k
views
I can't believe it's not Replacement!
(I feel like I might have to apologise in advance for this question, but oh well..)
I just rediscovered a comment from Asaf K here on MO that states that full Replacement is not needed for Borel ...
CommunityBot
- 1
6
votes
1
answer
587
views
Large cardinals in ZF + DC + AD
The Axiom of Dependent Choice (DC) is often considered to be an "intuitive and non-controversial" version of choice used in the proofs of many theorems in Analysis. Similarly, the Axiom of ...
- 4,713
8
votes
0
answers
169
views
Upper-bounding determinacy
While the converse of Borel determinacy ("If a set of reals is determined, then it is Borel") is boringly disprovable, I'm curious if there is a sense in which something like it is ...
- 21.1k
12
votes
1
answer
448
views
Comparing generic versions of $\mathbb{R}$
This question was previously asked and bountied at MSE, unsuccessfully.
I'm currently interested in the behavior of cardinalities in generic extensions of models of $\mathsf{ZF+\neg AC}$, and ...
- 2,520
5
votes
1
answer
150
views
Can these alternating series games be undetermined?
To each pair $(S,\mathcal{X})$ where $S=(s_i)_{i\in\mathbb{N}}$ is a decreasing sequence of positive real numbers and $\mathcal{X}\subseteq\mathbb{R}$, we can associate the alternation game $A_S(\...
- 4,697
9
votes
0
answers
251
views
Another determinacy-related cardinal characteristic
This question is a kind of "dual" to an earlier one of mine.
Although I don't know a reference for this, it's easy to show the following result:
Suppose $G$ is a game in which neither ...
- 21.1k
9
votes
0
answers
178
views
Does determinacy imply unravellability for the Borel sets (over a weak base theory)?
As far as I know, the only way we currently know how to prove Borel determinacy in $\mathsf{ZFC}$ is to go through unravelability (a rather technical property whose definition can be found in Martin's ...
- 21.1k
9
votes
0
answers
242
views
Is this cardinal characteristic trivial? (Number of strategies needed to guarantee at least one win)
(Previously asked at MSE.)
Let the determinacy number, $\mathfrak{g}$ (for "game"), be the smallest cardinal such that for every (two-player, perfect-information, length-$\omega$) game on $\...
- 21.1k
10
votes
0
answers
222
views
Is any choice axiom other than WISC inherited by Grothendieck topoi?
It is well known that even if one works with say ZFC as a base theory, Grothendieck topoi do not in general satisfy even fairly weak axioms like countable choice or small violations of choice and one ...
- 1,947
2
votes
0
answers
140
views
Weakening of open determinacy for uncountably long games
For a cardinal $\kappa$ I'll use the phrase "$(\kappa,\kappa)$-game" to mean "two-player, perfect-information, deterministic game on $\kappa$ of length $\kappa$."
Say that a ...
CommunityBot
- 1
11
votes
0
answers
374
views
A game of harmonic series(s)
Given a set $A\subseteq\mathbb{R}_{>0}$, consider the following (two-player, perfect-information, length-$\omega$) game $H_A$:
Players $1$ and $2$ alternately play strictly increasing natural ...
- 21.1k
2
votes
0
answers
100
views
Strong determinacy principles
It seems to be a well-known result that $\mathrm{AD}_{\omega_1}$ and $\mathrm{AD}_{\mathscr{P}(\mathbb{R})}$ (determinacy of $\omega$-length games with moves in $\omega_1$ or $\mathscr{P}(\mathbb{R})$,...
- 1,155
18
votes
1
answer
822
views
What sets can be unraveled?
A set $X\subseteq\omega^\omega$ is unravelable iff there is a possibly larger set $A$ and a clopen set $Y\subseteq A^\omega$ (with respect to the product topology coming from the discrete topology on $...
- 21.1k
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 ...
- 4,713
6
votes
1
answer
260
views
Existence of a winning strategy in the $\boldsymbol{\Delta}_2^0$ game
Consider the following infinite perfect information game with two players (the name I gave in the title of the post it totally made up): at each round $i \in \omega$, player $\mathrm{I}$ picks a ...
- 9,902
7
votes
1
answer
349
views
Uniformization under AD
Can the following uniformization statement be proved by $ZF+AD+DC$?
For any binary relation $R\subseteq \mathbb{R}^2$ with the property that $\forall x (\{y\mid R(x,y)\}\mbox{ is at most countable ...
- 9,902
3
votes
1
answer
153
views
Analytic sets and Turing determinacy
I wonder whether the following question have a positive answer within $ZFC$.
Question If $\{A_n\}_{n\in \omega}$ is a sequence of analytic sets so that $\bigcup_n A_n=2^{\omega}$, then there must be ...
- 3,073
7
votes
0
answers
231
views
Determinacy of symmetric games
Is it consistent that for all ordinals $α$ and $λ$ and infinite regular cardinals $κ$, games on $V_λ$ with game length $κα$ and $\mathrm{OD}(\mathrm{On}^κ)$ payoff that depends only on the set of all ...
- 7,545
12
votes
0
answers
505
views
Can a generic $\mathbb{R}$ have a new cardinality?
This question was asked and bountied at MSE, without success.
My main question is whether, starting with a model of determinacy, a "generic $\mathbb{R}$" could be different in cardinality ...
- 21.1k
14
votes
1
answer
840
views
Is there a minimal inner model for determinacy?
Assume $\sf ZF+AD$. Is there some inner model $M$ containing all the ordinals such that $M\models\sf ZF+AD$ as well?
What if we require $\omega_1$ and/or $\omega_2$ to be computed correctly?
Can we ...
- 8,099
2
votes
1
answer
203
views
Determinacy and polynomial time degrees
Is there a function $f:2^{<ℕ}→\{0,1\}$ such that for all $X∈{2^ℕ}$ with $X_{2i+1}=f(X_0,...,X_i)$ all hyperarithmetical properties of the polynomial time degree of $X$ are independent of $X$?
The ...
- 63.9k
3
votes
1
answer
301
views
Pointwise definable models of determinacy
Suppose that the Axiom of Determinacy (AD) holds in $L(ℝ)$, and for some statement φ, $α$ is minimal such that $L_α(ℝ)⊨φ$. Do definable (in $L_α(ℝ)$) elements of $L_α(ℝ)$ form an elementary ...
- 8,099
8
votes
1
answer
232
views
Aronszajn Trees when AC fails
This question may be easy and indicative of my ignorance about the failure of the axiom of choice. If so, I apologize. Below assume $\mathsf{DC}$ but not $\mathsf{AC}$. Suppose we have a partial order ...
- 39.7k
12
votes
0
answers
357
views
Undetermined copy/diagonalize games without CH
This question was the motivation behind an earlier question of mine; having thought about it some more, that question seems nontrivial and the connection is actually pretty tenuous anyways, so I've ...
- 21.1k
9
votes
0
answers
256
views
A bi-modal logic related to determinacy
The short version of my question is as follows. There is a natural (I hope!) way to associate a bimodal theory to a game (two-player, perfect-information, length-$\omega$, on $\omega$); are there "...
- 21.1k
2
votes
0
answers
196
views
An analytic game and Ramsey's theorem
Let $$P=\{(l_n, m_n)_{n=1}^t: l_n, m_n,t\in\mathbb{N}, l_1<\ldots <l_t, m_1<\ldots <m_t\}$$ and suppose that $T$ is some subset of $P$. Suppose that $T$ also has the following properties: ...
- 59
5
votes
0
answers
246
views
Forcing absoluteness in the setting of second-order arithmetic
There are some results about (set-size) forcing absoluteness for first-order properties in $L(\mathbb{R})$ and for $\mathbf{\Pi}^1_{\infty}$ properties when one works over $\mathsf{ZFC}$. My question ...
- 5,821
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 ...
- 236k