All Questions
22 questions
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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 ...
5
votes
1
answer
487
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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
7
votes
1
answer
443
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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) ...
- 959
2
votes
0
answers
235
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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
4
votes
1
answer
268
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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 ...
- 959
4
votes
0
answers
253
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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 $\kappa$, ...
- 1,490
4
votes
1
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469
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How large are the stabilization times of Ordinal Turing Machines with an oracle for the transfinite initial ordinals?
This question is based on the assumption that $V \ne L$ and we have $\omega_1^L < \omega_1$ (here $\omega_1^L$ is equal to the supremum of ordinals accidentally writable by no-oracle Ordinal Turing ...
- 959
4
votes
3
answers
403
views
Countably infinite sets of ordinals as parameters for Ordinal Turing Machines
Let $A$ and $B$ denote two countably infinite sets of ordinals.
Let $W_A$ denote the supremum of ordinals writable by Ordinal Turing Machines with the set $A$ given as the source of parameters. That ...
- 959
4
votes
1
answer
227
views
Is there a real $x$ which is eventually writable from an ordinal parameter $\alpha < \omega_1$, but not from $\omega_1$?
According to Lemma 3.14 in the paper “Recognizable sets and Woodin cardinals: Computation beyond the constructible universe”, there is a real $x$ in $L$ which is recognizable from some ordinal $\alpha$...
- 959
4
votes
1
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337
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How to compare three supremums of ordinals eventually writable by Ordinal Turing Machines?
This question implies that we have fixed: (i) a particular enumeration of Ordinal Turing machines; (ii) a particular way to encode an ordinal by an infinite binary sequence.
The class of $[1]$-...
- 959
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 ...
- 959
1
vote
1
answer
287
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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 ...
- 959
5
votes
1
answer
749
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Which ordinal is larger, the supremum of ordinals writable by iterated Infinite Time Turing Machines or the smallest $\Sigma_2^1$-reflecting ordinal?
The ordinal $\tau_1$ corresponds to $\lambda^{\textit{it}}$ (the supremum of all ordinals writable by iterated ITTMs) — see Definition 3.1 in the paper “ITTMs with Feedback” [Robert S. Lubarsky]. ...
- 959
6
votes
0
answers
303
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Weaker versions of Gandy ordinals
Gostanian's paper "The next admissible ordinal" (see https://www.sciencedirect.com/science/article/pii/0003484379900251 ), is concerned with the supremum of the $\alpha$-recursive ordinals for various ...
- 365
9
votes
1
answer
711
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Computable models of the ordinal numbers
It's known, for example in the answer to this question: Is there a computable model of ZFC? that ZFC has no computable model. My questions is: is there a model of ZFC for which the order relation on ...
- 375
4
votes
0
answers
199
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On the proof of a normal form theorem for ordinal (primitive) recursion
Consider the following statement (which follows easily from various results found in the literature):
(†) There exists a primitive recursive (“p.r.”) relation $T$ on the ordinals such that, if $(...
- 32.5k
4
votes
2
answers
489
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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 ...
- 881
13
votes
1
answer
650
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About primitively recursively recognizable ordinals
Preliminary: I believe the notion of primitive recursive functions on ordinals is standard and unproblematic (the main difference with the finite case is that one needs to introduce a $\sup$ or $\...
- 32.5k
5
votes
1
answer
627
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On a characterization of the recursively inaccessible ordinals
For a given set of numbers $A$, let $O^A$ be the hyperjump of $A$. It is possible to iterate inductively the hyperjump of a set, through the computable ordinals, in a way that the $\alpha$-th ...
- 764
27
votes
1
answer
2k
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Why isn't this a computable description of the ordinal of ZF?
In a previous MO question, I was told by several commenters that
(a) it's known that there exists a computable ordinal $\alpha_{ZF}$ that "encodes the strength of ZF set theory" (i.e., a least ...
- 9,833
17
votes
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-...
- 2,992