All Questions
Tagged with computability-theory set-theory
172 questions
5
votes
0
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158
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If $\omega_1$ is not inaccessible in $L$, how hard can it be to find a non-measurable $\Sigma^1_3$ set of reals?
In his wonderfully titled paper Can you take Solovay's inaccessible away? Shelah showed that if every $\mathbf{\Sigma}^1_3$ set of reals is Lebesgue measurable, then $\omega_1$ is an inaccessible ...
- 12.5k
2
votes
1
answer
161
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Are Cohen Generics Minimal Covers?
Are Cohen generics (in $2^\omega$) minimal covers?
I'm ultimately interested in this question for some more effective notion of forcing but I realized I wasn't sure how to show this even assuming full ...
- 44.4k
4
votes
2
answers
489
views
Mapping between Notations
$\DeclareMathOperator{\address}{address}$
As in my other question, it is assumed that the (total) function describing a given notation is denoted as $\address:p \rightarrow \Bbb{N}$ and assumed to be ...
- 6,367
6
votes
2
answers
276
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Extending polynomial hierarchy above $\omega$
The arithmetic hierarchy is naturally extended to all ordinals via ordinal notations creating a hierarchy for all hyperarithmetic sets. The polynomial time hierarchy is defined analogously to the ...
- 328
6
votes
0
answers
298
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What are these non-classical versions of ZFC defined by realizability?
See Kleene realizability in Peano arithmetic for a similar question, but about PA instead of ZFC. (In particular, an answer as specific as Emil Jeřábek's answer would be great!)
In the context of ...
- 6,353
18
votes
2
answers
1k
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Is there a name for sets for which it is easier to test membership than to find members---and vice versa?
This is a question my son Bob asked me. For some sets it is relatively easy
to test for membership but a lot more difficult to find members, and for others
the reverse is true. Here is an elementary ...
- 236k
4
votes
0
answers
149
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Computable subsets of non-standard models of arithmetic
By Tennenbaum's theorem, there exists no computable non-standard model of $\mathsf{PA}$. That is, for any nonstandard model $M$, we cannot define an encoding of the integers $x\in M$ such that $+_M$, $...
- 291
17
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7
answers
2k
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Finding the largest integer describable with a string of symbols of predefined length
(This question is motivated by the reading of the article Large numbers and unprovable theorems by Joel Spencer, which can be found at http://mathdl.maa.org/images/upload_library/22/Ford/Spencer669-...
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33
votes
15
answers
7k
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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&...
- 103
15
votes
2
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918
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Which are the hereditarily computably enumerable sets?
My question is about sets that are computably enumerable with respect to their hereditary membership structure. Specifically, let me define that a hereditarily computably enumerable (h.c.e.) set is ...
- 7,545
1
vote
0
answers
123
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Is possibile to define transfinite sum and product recursively? [closed]
On mathstackexchange a few days ago I published the following question where I asked about "transfinite" sum and products but actually nobody answered or gave an opinion with a comment: thus ...
6
votes
1
answer
571
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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 ...
- 2,031
2
votes
0
answers
118
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Uniformization and functions on Turing degrees
Assuming Martin's Conjecture on functions between Turing degrees, is AD + DC consistent with existence of an $f:\mathcal{D}_t → \mathcal{D}_t$ of rank $Θ$ ?
$\mathcal{D}_t$ is the set of Turing ...
- 7,545
9
votes
0
answers
471
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(A little bit) Beyond the E-recursive
The E-recursive functions are a particular generalization of classical recursion theory to the entire set-theoretic universe, $V$. They are defined via a schemes: see Sacks' $E$-recursive intuitions. ...
CommunityBot
- 1
7
votes
2
answers
644
views
Ideals generated by Turing independent sets
Recall that $X \subseteq 2^{\omega}$ is Turing independent if no $y \in X$ is computable from the Turing join of any finite subset of $X \setminus \{y\}$.
Question 1. Can we construct a Turing ...
- 1,780
3
votes
0
answers
143
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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
3
votes
2
answers
262
views
Question regarding $W$ as not hyperarithmetic
Consider the indexes of all ordinary programs generating functions from $\mathbb{N}^2$ to $\{0,1\}$. If we let $W$ be the set of exactly of all those indexes $e$ such that $\phi_e$ computes a total ...
- 21.2k
4
votes
3
answers
406
views
Hyperarithmetically least elements in $\Pi^1_1$ sets
My question is: Do we have a hyperarithmetically $\le_H$-least real in any $\Pi^1_1$ set? That is
Question. Suppose that $A$ is a non-empty $\Pi^1_1$ set. Then can we find a real $a\in A$ such that $...
CommunityBot
- 1
8
votes
2
answers
518
views
History of forcing over admissible sets
In his paper "Forcing in admissible sets", Ershov writes
In unpublished lectures given at Novosibirsk State University in 1976-1977 on the theory of admissible sets, the author
showed that it is ...
- 17.7k
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 ...
- 5,031
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}<_{...
- 2,031
2
votes
1
answer
135
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Harrington's notes on McLaughlin/Arithmetically incomparable singletons
At one point I had copies of the handwritten notes Leo created about the McLaughlin conjecture and I know a similar set of notes exist titled Arithmetically incomparable arithmetic singletons. I've ...
- 3,029
45
votes
5
answers
64k
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How large is TREE(3)?
Friedman, in _Lectures notes on enormous integers shows that TREE(3) is much larger than n(4), itself bounded below by $A^{A(187195)}(3)$ (where $A$ is the Ackerman function and exponentiation ...
- 257
4
votes
1
answer
172
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Let $\pi$ be a $ℍ𝑌𝑃_𝔐$-recursive projection of $ℍ𝑌𝑃_𝔐$ into 𝔐. What does $ℍ𝑌𝑃_{(𝔐, Domain(\pi))}$ contain?
Let the structure $\mathfrak{A} = (A, R_1, ..., R_n)$ be strongly acceptable iff $\mathfrak{A}$ is an acceptable structure (in the sense of Moschovakis' Elementary Induction on Abstract Structures), $\...
- 43
1
vote
1
answer
362
views
How are Koepke's ordinal computability and E-recursion related?
In Koepke's paper, "Turing Computations On Ordinals", one has the following (well-known) result:
A set $x$ is ordinal computable from a finite set of ordinal parameters if and only if it is ...
- 3,065
2
votes
0
answers
133
views
Higher-order oracle computation of reals and axiom of constructibility
Certain real numbers can be approximated arbitrarily well by computable functions. If we introduce halting oracles, then more real numbers can be "computed", like Chaitin's constant or the ...
- 173
4
votes
0
answers
182
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Some questions on a paper of Gerald Sacks
I've been reading Sacks' Countable admissible ordinals and hyperdegrees as I'm interested in Theorem 5.3 of the paper:
Let $M$ be a countable standard model of $\mathsf{ZF}$ and $V=L$. Suppose $\...
- 2,286
7
votes
1
answer
443
views
Gaps in the ordinals writable by Ordinal Turing Machines with a single countable parameter
Let $W(\alpha)$ denote the set of all (countable) ordinals writable by Ordinal Turing Machines with a single (countable) parameter $\alpha$, i.e. each computation starts with a single ($\alpha$-th) ...
- 236k
18
votes
4
answers
2k
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Theorems in set theory that use computability theory tools, and vice versa
I recently learnt that the proof of the classical theorem "$\mathsf{AD}$ $\implies$ $\aleph_1$ is measurable" uses computability theory tools (or at least one of its proofs does so). I'm ...
- 4,201
2
votes
1
answer
176
views
How can Kőnig's Lemma be expressed in Monadic Second-Order Logic of 2 Successors?
I read the following on Wikipedia's page on Monadic Second-Order Logic of Two Successors (MS2S):
Weak S2S (WS2S) requires all sets to be finite (note that finiteness
is expressible in S2S using Kőnig'...
- 21.2k
1
vote
1
answer
287
views
How large is the smallest ordinal larger than any “minimal ordinal parameter” for any pair of an Ordinal Turing Machine and a real?
In this question, the notation $P^x(\alpha)$ denotes a situation where a particular OTM-program $P$ performs a computation on input $x$ with an ordinal parameter $\alpha$, assuming that $x$ is written ...
- 3,042
2
votes
1
answer
202
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A question about computability and Turing machines Part 2
I asked a question a few days ago and got a response
But my follow-up question was not answered (maybe my email was not sent successfully)
A question about computability and Turing machines
My quesion ...
- 236k
2
votes
1
answer
298
views
A question about computability and Turing machines
For any recursively enumerable set theory $T$ (of consistency strength at least superior to KP), if we want to calculate $F(n)=\{F(m)：m∈ω∧mEn\}$ and can determine each $F(n)$ for a Henkin model $(ω,E)$...
- 32.5k
3
votes
0
answers
110
views
What is the $E$-r.e. part of $L$?
See Sacks' paper $E$-recursive inuitions or his book for background on $E$-recursion. Throughout, work in $\mathsf{ZFC+V\not=L}$. I'll use $\varphi_e$ in place of $\{e\}$ for the $e$th partial $E$-...
- 21.2k
6
votes
0
answers
117
views
Reverse mathematics of Banach-Mazur games
Given $\mathcal{A}\subseteq\omega^\omega$, the Banach-Mazur game with payoff set $\mathcal{A}$ consists of players $1$ and $2$ alternately playing nonempty finite strings of naturals with player $1$ ...
- 21.2k
6
votes
0
answers
151
views
Complexity of constructive arithmetical truth vs second order arithmetic
Let us say that an arithmetic statement is constructively true iff it is realized by a computable function under Kleene's function realizability. Does the set of constructively true (first order) ...
- 7,545
2
votes
1
answer
154
views
Axiomatization of S2S
What is a reasonable axiomatization of S2S?
S2S is the monadic second order theory with two successors (Wikipedia link). It has finite binary strings, operations $s→s0$ and $s→s1$ on strings, and ...
- 7,545
4
votes
0
answers
253
views
Is this recursion theoretic analogue of a criterion of weakly compact cardinal accurate?
Jensen proved that, if V=L, and $\kappa$ is a regular cardinal, then if for any stationary $A\subseteq \kappa$, the set $\{\alpha\mid A \text{ is stationary below }\alpha\}$ is stationary in $\kappa$, ...
- 2,031
6
votes
1
answer
201
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Reference request: generalized randomness
There is a plethora of notions of randomness for Cantor space ($\phantom{}^\omega 2$) (Schnorr randomness, Martin-Löf randomness, weak 2-randomness, the various forms of higher randomness such as $\...
- 14k
2
votes
0
answers
235
views
The supremum of ordinals eventually writable by Ordinal Turing Machines with an oracle for the class of stabilization ordinals
This question is based on the assumption that all computations start with no ordinal parameters (i.e. the input is empty).
The term “stabilization time of a machine” for this question implies the ...
- 959
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
4
votes
1
answer
268
views
Existence of a particular function that maps an arbitrary set of ordinals to a single ordinal
Does there exist a function $f$ that satisfies all of the following three properties?
The function converts an arbitrarily large (empty, finite, countably/uncountably infinite) set of ordinals to a ...
- 17.7k
14
votes
2
answers
1k
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Does Turing determinacy imply full determinacy?
The axiom of Turing determinacy is a weakening of the full axiom of determinacy, $AD$, in which only games with payoff sets which are $\equiv_T$-invariant are demanded to be determined.
In "...
- 32.5k
6
votes
0
answers
806
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A strong plus-one hypothesis
To make this more easily readable, I'll start with the question and then give the explanation/motivation.
Question. Is the following principle (or its weakening, with "for every real $r$" ...
- 35.5k
4
votes
3
answers
495
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How large is the supremum of halting times of Infinite Time Turing Machines, assuming that halting times are bounded and inputs are arbitrary?
Given a fixed enumeration of Infinite Time Turing Machines (ITTMs), let $M_i(x)$ denote a computation of an $i$-th ITTM, assuming that the input $x$ is a real (an infinite binary sequence).
Then the ...
- 1
5
votes
1
answer
286
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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, \...
- 4,713
6
votes
1
answer
227
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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{...
- 17.7k
7
votes
0
answers
313
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An uncountable structure with unusual "relatively-computable shadow"
Below, all structures are infinite and in a finite language. Given a structure $\mathcal{A}$ with domain $\omega$, we conflate $\mathcal{A}$ with some reasonable encoding of its atomic diagram for ...
- 21.2k
1
vote
1
answer
260
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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 ...
CommunityBot
- 1
6
votes
0
answers
207
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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