All Questions
22 questions
6
votes
0
answers
179
views
$Δ^1_3$ reals in transitive models
Every real number is $Δ^1_2$ in some $ω$-model of ZFC\P. I am looking for analogous statements for $Δ^1_3$ definability and transitive models, where the situation is more subtle.
What is the ...
- 7,545
3
votes
0
answers
240
views
The most powerful inner model and a $\Delta^2_1$ well-ordering of the reals
With the current research, it seems that we are in a position to get extremely powerful absoluteness theorems (like $\Sigma^2_0$-absoluteness, $\Sigma^2_1$-absoluteness, $\Sigma^2_2$, $\diamondsuit_G$,...
- 695
2
votes
1
answer
269
views
Inner model for KP and a Well-Ordering of the Reals
It is well known that Gödel proved the following theorem:
$\mathsf{ZFC + V=L}$ has a $\mathit{\Delta}^1_2$-good well-ordering of $\mathbb{R}$. (Gödel, Addison)
So:
Is there an inner model for KP/Z/....
- 695
5
votes
1
answer
243
views
Inner model with a $\mathit{\Delta}^1_3$-good well-ordering of the reals
The constructible universe $L$ has some nice properties:
$L$ has a $\mathit{\Delta}^1_2$-good well-ordering of $\mathbb{R}$. (Gödel, Addison)
For any $\mathit{\Sigma}^1_2$ formula $\varphi(x)$ and a ...
- 2,286
7
votes
1
answer
222
views
Does $\mathit{Aut}(\mathbb{R};+)$ have a copy in $L(\mathbb{R})$ granting large cardinals?
Throughout, work in $\mathsf{ZFC}$ + large cardinals (let's say a proper class of Woodin limits of Woodins but I'm happy to go higher if that would help).
Let $\mathcal{R}=(\mathbb{R};+)$ be the ...
- 20.9k
3
votes
0
answers
152
views
Why are the sharps of sets of big ordinals not in $\mathcal{P}(\omega)$?
In his talk A Condensed History of Condensation, Welch presents the following recursive sharp function, that is total when all sharps exist:
\begin{align*}
\# \colon ON &\to \mathcal{P}(ON) \\
\...
- 421
3
votes
1
answer
241
views
Do all limit $\alpha \in \omega_1^L$ satisfy $L_\alpha \models V=HC$?
In Gaps in the constructible universe, Marek and Srebrny, 1973 a gap ordinal and the start of a gap are defined as follows
$\alpha$ is a gap ordinal iff $(L_{\alpha+1}-L_\alpha)\bigcap \mathcal{P}(\...
- 421
2
votes
1
answer
255
views
Why can't $L_\beta$ contain a real coding a well-ordering of order-type $\beta$, when $\beta$ is a gap ordinal?
In Gaps in the constructible universe, Marek and Srebrny, 1973 a gap ordinal is defined as follows
$\alpha$ is a gap ordinal iff $(L_{\alpha+1}-L_\alpha)\cap \mathcal{P}(\omega) = \emptyset$
Their ...
- 421
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 $...
- 20.9k
3
votes
1
answer
302
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 ...
- 7,545
12
votes
1
answer
464
views
Does $Π^1_{2n+1} = (Σ^1_{2n+1})^{M_{2n-1}}$?
Every $Π^1_1$ formula $φ$ without free second order variables can be converted into a $Σ^1_1$ $ψ$ such that $φ ⇔ ψ^\mathrm{HYP}$, and vice versa. ($\mathrm{HYP}$ is the hyperarithmetical universe, ...
- 7,545
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" ...
- 20.9k
6
votes
1
answer
385
views
Absoluteness for the Chang model
Often, large cardinals imply that many definable inner models and transitive sets have theories which are absolute under forcing. For example, assuming a proper class of Woodin cardinals, the theory ...
- 20.9k
2
votes
0
answers
149
views
Strong determinacy principles for "small" sets
In the course of a (fun but silly) project I'm working on, I've run into the following class of determinacy notions:
Suppose I have a "reasonably definable" class $\mathcal{C}$ of games (in the usual ...
- 20.9k
4
votes
0
answers
236
views
Does absoluteness imply a club dichotomy?
My question is about two types of consequence of large cardinals, considered over ZFC on their own.
First, we have statements of the form, "The club filter on $\omega_1$ is an ultrafilter when ...
- 20.9k
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, ...
- 20.9k
7
votes
1
answer
313
views
On thin $\Sigma^1_2$ equivalence relations
This question is regarding Hjorth's paper "Some applications of coarse inner model theory", J. Symbolic Logic 62 (1997), no. 2, 337–365.
Hjorth claims that if $E$ is a thin $\Sigma^1_2$ equivalence ...
- 685
8
votes
4
answers
906
views
Is it inconsistent for a model of set theory to contain its own first order theory?
I am wondering if it is inconsistent to have a model of set theory V such that V contains an $A\subset \omega$ that codes its first order theory.I.e. for all $\{\underline\epsilon\}$-sentences $\phi$, ...
- 603
4
votes
0
answers
142
views
Characterization of $L[T_{2n+1}]$ as a direct limit of mice
I am asking for a reference request/proof sketch for the result of Steel
that characterizes $L[T_{2n+1}]$ as a direct limit of mice.
Given that both $L[T_{2n+1}]$ and $M_{2n}$ have a $\Sigma_{2n+2}$ ...
- 603
7
votes
2
answers
657
views
Topological tameness beyond the Gandy-Harrington topology
The Gandy-Harrington topology on $\omega^\omega$ is the topology generated by all lightface $\Sigma^1_1$ sets; that is, all sets which are continuous-in-the-usual-sense images of $\omega^\omega$.
...
- 20.9k
7
votes
1
answer
344
views
Characterizing L(R) Cardinals in HOD
We're working in L(R) under AD.
We know that
$\omega_1$ is the least measurable in HOD, $\Theta$ is the least woodin, $\delta^2_1$ is the least strong to the woodin, etc.
My question is about ...
- 603
5
votes
1
answer
622
views
Models of Determinacy
Today we have that $L(\mathbb{R}) \models AD$ (assuming there are $\omega$ many Woodin cardinals and a measurable above them all). I was wondering what other models of $AD$ might look like and if it ...
- 3,804