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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 ...
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 ...
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\...
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 ...
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 (...
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 ...
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 ...
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) \\ \...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $\varphi$ ...
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 ...