All Questions
19 questions
3
votes
0
answers
143
views
Lindström's theorem part 2 for non-relativizing logics
By "logic" I mean the definition gotten by removing the relativization property from "regular logic" — see e.g. Ebbinghaus/Flum/Thomas — and adding the condition that for every ...
- 21.2k
10
votes
2
answers
470
views
Is the set of permissible numbers of models of various cardinalities computable?
This question arose in the comments to this question.
Let $X$ be the set of pairs $(m,k)$ such that there is some (consistent complete countable first-order) theory $T$ with exactly $m$ models of size ...
- 21.2k
5
votes
1
answer
487
views
How to solve this exercise about large countable ordinals?
In this note (Notes on Higher Type ITTM-recursion, 2021) written by Philip Welch, I'm trying to solve exercise 3.5(i), but I don't know how to solve it.
The problem is: assume that $L_{\gamma_0}<_{...
- 1,490
6
votes
1
answer
571
views
Parameter-free effective cardinals
In the paper "Effective cardinals and determinacy in third order arithmetic" by Juan Aguilera, effective cardinals is defined.
I'm curious about its little variation, parameter-free ...
- 1,490
6
votes
1
answer
227
views
How often are forcing extensions of countable computably saturated models of $\mathsf{ZFC}$ computably saturated?
Recall that given a finite language $\mathcal{L}$, we say that an $\mathcal{L}$-structure is computably saturated (or recursively saturated) if for any computable set $\Sigma(\bar{x},y)$ of $\mathcal{...
- 12.5k
6
votes
0
answers
207
views
Fragments of infinitary logic with a weak definability property
For a countable admissible ordinal $\alpha$, let $\mathcal{L}_\alpha=\mathcal{L}_{\infty,\omega}\cap L_\alpha$ and let $\equiv_\alpha$ be the corresponding elementary equivalence relation. Say that ...
- 21.2k
1
vote
1
answer
260
views
Natural strong logic with Barwise compactness property
Throughout, by "logic" I mean regular logic (in the sense of Ebbinghaus–Flum–Thomas) whose sentences are coded by elements of $\mathsf{HC}$. Say that $\mathcal{L}$ is Barwise compact iff ...
- 21.2k
5
votes
0
answers
317
views
$\Sigma_n$-complete sets in the Levy hierarchy
Recall that a set $A \subseteq \mathbb N$ is (many-one, Turing) $\Sigma_n$-complete if it's $\Sigma_n$ and any other $\Sigma_n$ set (many-one, Turing) reduces to it. This definition actually makes ...
- 1,882
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 ...
- 21.2k
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 ...
- 21.2k
9
votes
0
answers
279
views
What logic characterizes relative intrinsic complexity in set recursion?
Short version: Is there an analogue of the Ash-Knight-Manasse-Slaman/Chisholm theorem for $E$-recursion?
Long version: I'm interested in "$E$-recursive structure theory," but it's not ...
- 21.2k
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 ...
- 21.2k
6
votes
0
answers
249
views
Number of models vs. complexity for SOL theories
This was previously asked at MSE without success.
Suppose $T$ is a complete first-order theory with continuum-many countable models up to isomorphism. We define two sets of Turing degrees associated ...
- 21.2k
17
votes
1
answer
720
views
Would an oracle for Rayo's function let you compute a model of $(V, \in)$?
Working in Kelly-morse set theory, let $R$ be an oracle that can compute Rayo's function. Can $R$ compute a countable model $M = (\mathbb N,\in_M)$ that is elementary equivalent to $(V, \in)$?
- 6,353
5
votes
1
answer
286
views
Is there an oracle that can compute something iff it is computable in every countable model that is equivalent to $(V, \in)$?
Let us work in Kelly-morse set theory, so we can talk about $V$. For some model $M=(\mathbb N, \in_M)$ that is elementary equivalent $(V, \in)$, we can have an oracle that corresponds to $(\mathbb N, \...
- 6,353
13
votes
1
answer
742
views
Is forcing computable?
By results similar to Tennenbaum's theorem we know that there exist no computable models of $ZF$. But suppose we are given, as a sort of oracle, access to some model of $ZF$ (e.g. we can make oracle ...
- 28.2k
4
votes
1
answer
677
views
Are there sets which are computable in one model, but uncomputable in another?
Suppose we have two models of set theory, $U$ and $V$ which have the same $\Bbb N$. Is it possible that there is a set $A\subseteq\Bbb N$ such that, in $U$, this set is computable, i.e. there is a ...
- 28.2k
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
33
votes
15
answers
7k
views
What's a magical theorem in logic?
Some theorems are magical: their hypotheses are easy to meet, and when invoked (as lemmas) in the midst of an otherwise routine proof, they deliver the desired conclusion more or less straightaway&...