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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 ...
Noah Schweber's user avatar
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 ...
Noah Schweber's user avatar
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 ...
Noah Schweber's user avatar
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 ...
saolof's user avatar
  • 1,947
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 ...
Trevor Wilson's user avatar
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 ...
Noah Schweber's user avatar
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 ...
Noah Schweber's user avatar
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 $\...
Noah Schweber's user avatar
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 "...
Noah Schweber's user avatar
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 ...
Asaf Karagila's user avatar
  • 39.7k
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 ...
Noah Schweber's user avatar
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 ...
Dmytro Taranovsky's user avatar
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 ...
Tim Campion's user avatar
  • 63.9k
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^{<\...
Noah Schweber's user avatar
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 ...
Dmytro Taranovsky's user avatar
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}^{...
Trevor Wilson's user avatar
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 ...
Dmytro Taranovsky's user avatar
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 ...
Dmytro Taranovsky's user avatar
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 ...
Fedor Pakhomov's user avatar
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 ...
Noah Schweber's user avatar
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 $\alpha$ ...
John Gowers's user avatar
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 ...
Dmytro Taranovsky's user avatar
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 ...
Dmytro Taranovsky's user avatar
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(\...
Obrad Kasum's user avatar
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 ...
Dmytro Taranovsky's user avatar
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). ...
Dmytro Taranovsky's user avatar
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 ...
Dmytro Taranovsky's user avatar
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 ...
Noah Schweber's user avatar
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 ...
Noah Schweber's user avatar
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 $\mathscr{P}(\mathbb{R})$,...
Beau Madison Mount's user avatar
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: ...
user-1's user avatar
  • 59
2 votes
0 answers
1k views

The Rise and Fall of Dictators & How it Depends on Our Choice

This question is loosely inspired by the following paper of Shelah on Arrow property in which he answered a question of Gil Kalai affirmatively. Shelah, Saharon, On the Arrow property. Adv. in Appl. ...
Morteza Azad's user avatar
0 votes
0 answers
160 views

Strength of $Δ^1_{2n}$ determinacy

According to Lightface mice with finitely many Woodin cardinals from optimal determinacy hypotheses by Yizheng Zhu, theorem 1.1, over $\mathbf{Σ^1_{2n+1}}$ determinacy, $Δ^1_{2n+2}$ determinacy is ...
Dmytro Taranovsky's user avatar