All Questions
16 questions
25
votes
2
answers
1k
views
Detecting uncountable cardinals in $(\mathbb{R};+,\times,\mathbb{N})$
For a structure $\mathcal{X}=(X;...)$, say that a cardinal $\kappa$ is $\mathcal{X}$-detectable iff there is some sentence $\varphi$ in the language of $\mathcal{X}$ together with a fresh unary ...
- 20.5k
18
votes
3
answers
1k
views
How badly does compactness fail in $\mathcal{L}_{\omega_1\omega}$?
I would like to get a better idea of how badly compactness fails in $\mathcal{L}_{\omega_1\omega}$.
Let $\Gamma$ be an arbitrary set of sentences from $\mathcal{L}_{\omega_1\omega}$. Let the ...
- 1,142
16
votes
6
answers
2k
views
Application of Fraïssé construction in set theory
As you know Fraïssé limit construction and its generalization, Hrushovski's construction, have many applications in model theory to build models with interesting property.
Now I would like to know ...
- 1,872
9
votes
1
answer
495
views
Can two versions of $\omega_1^{CK}(\mathsf{Ord})$ ever coincide?
The goal of this question is to fill in the gap in this old answer of mine.
For a transitive set $M$, thought of as an $\{\in\}$-structure, we define the following ordinals (this is not the notation ...
- 20.5k
9
votes
0
answers
237
views
Continuum hypothesis analogue for substructures
This question was previously asked and bountied at MSE. Throughout, "theory" means "possibly-incomplete first-order theory in a countable language."
Say that a theory $T$ has CHS (...
- 20.5k
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 ...
- 20.5k
8
votes
1
answer
514
views
How big is the least non-$\Sigma^1_1$-pointwise-definable ordinal?
There's a large countable ordinal which has cropped up (as a lower bound!) in a computable structure theory problem I'm playing with. At present I don't really understand how big it is, and I'm ...
- 20.5k
7
votes
1
answer
363
views
Is the consistency of $\mathcal{L}_{\infty\omega}$-sentences absolute?
The question is exactly that of the title. Suppose $\varphi\in V$ is an $\mathcal{L}_{\infty\omega}$-sentence, and $W$ is an inner model of $V$ such that $\varphi\in W$. Is the statement
$\varphi$ ...
- 20.5k
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 ...
- 20.5k
7
votes
0
answers
304
views
Which countable ordinals are "Barwise compact" for $\mathcal{L}_{\infty,\omega_1}$?
Barwise compactness says (as a special case) that whenever $\alpha$ is countable and admissible, $T\subseteq\mathcal{L}_{\infty,\omega}\cap L_\alpha$ is $\alpha$-c.e., and every subset of $T$ which is ...
- 20.5k
6
votes
0
answers
128
views
Detecting uncountable cardinalities, this time with determinacy
By "small cardinality" I mean a Scott cardinality onto which $\mathbb{R}$ surjects ($0$ isn't interesting here). $\mathcal{R}=(\mathbb{R};+,\times,\mathbb{Z})$ is the field of real numbers ...
- 20.5k
6
votes
0
answers
232
views
The number of countable models with determinacy
Throughout, work in $\mathsf{ZF+DC+AD_\mathbb{R}}$.
Given a theory $T$, let $[T]$ be the set of isomorphism types of models of $T$ with domain $\subseteq\omega$. This question is an outgrowth of this ...
- 20.5k
5
votes
1
answer
878
views
Countable admissible ordinals
Jensen claimed that for any finite increasing sequence countable admissible ordinals $\omega= \alpha_0<\alpha_1\cdots <\alpha_n$, there is a real $x$ so that, for each $m\leq n$, $\alpha_m$ is ...
- 4,201
5
votes
0
answers
218
views
Improving a Lindstrom-y fact about $\mathcal{L}_{\omega_1,\omega}$?
See e.g. the last section of Ebbinghaus/Flum/Thomas for the relevant background on abstract model theory. Below, all languages are finite for simplicity. "$HC$" is the set of hereditarily ...
- 20.5k
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
1
vote
0
answers
143
views
Is this approximation to infinitary equivalence coarse on countable structures?
This question is a kind of dual to this earlier one. Note that if we replace $\mathsf{FOL}$ with $\mathcal{L}_{\omega_1,\omega}$, things trivialize since we can use the theory $\{\varphi^A\...
- 20.5k