All Questions
Tagged with model-theory descriptive-set-theory
31
questions
3
votes
0
answers
168
views
Can the set of parafinite congruences be descriptive-set-theoretically complicated?
Fix an algebra $\mathfrak{A}$ with underlying set $\mathbb{N}$ and finite language $\Sigma$. The set of congruences on $\mathfrak{A}$ is a closed subset $C_\mathfrak{A}$ of $2^\mathbb{N}$ (with the ...
- 20.7k
14
votes
0
answers
419
views
Which functions have all the common $\forall\exists$-properties of continuous functions?
This is an attempt at partial progress towards this question. Meanwhile, Sam Sanders pointed out that my original term was already in use, as were a couple other back-up terms, so ... oh well.
For a ...
- 20.7k
32
votes
2
answers
2k
views
Quantifier complexity of the definition of continuity of functions
This was previously asked at MSE, but I was told to ask it on MO. Consider the structure $(\mathbb{R};+,-,*,0,1,<)$. We adjoin to it a unary function $f$ defined everywhere on the set of real ...
- 2,063
6
votes
0
answers
120
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 ...
- 20.7k
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\...
- 20.7k
7
votes
1
answer
208
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 ...
- 20.7k
9
votes
0
answers
229
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 (...
- 20.7k
8
votes
0
answers
167
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 ...
- 20.7k
6
votes
0
answers
222
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 ...
- 20.7k
3
votes
0
answers
141
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
7
votes
0
answers
296
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 ...
- 20.7k
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 ...
- 20.7k
3
votes
1
answer
125
views
Roelcke precompactness and Ramsey property
A survey by Nguyen Van Thé (2014) has Conjecture 1,
which is that
"every closed oligomorphic
subgroup of $S_∞$ should have a metrizable universal minimal flow with a generic
orbit." Later, ...
- 91
8
votes
1
answer
463
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 ...
- 20.7k
9
votes
1
answer
442
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 ...
- 20.7k