Some Questions in Computable Mathematics

Hirschfeldt, Denis R.

doi:10.1007/978-3-319-50062-1_4

Cited by 6 publications

(3 citation statements)

References 89 publications

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“…This notion is not really new. Hirschfeldt and Khoussainov (see, e.g., [19]) observed that every locally finite connected graph is PA(0 ′ )-categorical. We note that, building on a work of Gromov [17], Melnikov and Nies [28] obtained a similar bound for compact separable Polish spaces.…”

Section: The Resultsmentioning

confidence: 99%

Punctual Categoricity and Universality

Downey¹,

Greenberg²,

Melnikov³

et al. 2020

J. symb. log.

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We describe punctual categoricity in several natural classes, including binary relational structures and mono-unary functional structures. We prove that every punctually categorical structure in a finite unary language is ${\text {PA}}(0')$-categorical, and we show that this upper bound is tight. We also construct an example of a punctually categorical structure whose degree of categoricity is $0''$. We also prove that, with a bit of work, the latter result can be pushed beyond $\Delta ^1_1$, thus showing that punctually categorical structures can possess arbitrarily complex automorphism orbits.As a consequence, it follows that binary relational structures and unary structures are not universal with respect to primitive recursive interpretations; equivalently, in these classes every rich enough interpretation technique must necessarily involve unbounded existential quantification or infinite disjunction. In contrast, it is well-known that both classes are universal for Turing computability.

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Section: The Resultsmentioning

confidence: 99%

Punctual Categoricity and Universality

Downey¹,

Greenberg²,

Melnikov³

et al. 2020

J. symb. log.

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“…In this paper, we study the reverse mathematics of variants of this statement by considering various classes of non-computable functions over N → N. This study is motivated by the following question of Dzhafarov and Schweber in MathOverflow [6] and taken up by Hirschfeldt in his open questions paper [7]. A set X is a limit non-decreasing subsequence for a stable function f : N × N → N if it is a non-decreasing subsequence of its limit function f : N → N defined by f (x) = lim s f (x, s).…”

Section: Introductionmentioning

confidence: 99%

The reverse mathematics of non-decreasing subsequences

Patey

2017

Arch. Math. Logic

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Every function over the natural numbers has an infinite subdomain on which the function is non-decreasing. Motivated by a question of Dzhafarov and Schweber, we study the reverse mathematics of variants of this statement. It turns out that this statement restricted to computably bounded functions is computationally weak and does not imply the existence of the halting set. On the other hand, we prove that it is not a consequence of Ramsey's theorem for pairs. This statement can therefore be seen as an arguably natural principle between the arithmetic comprehension axiom and stable Ramsey's theorem for pairs.

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“…Question 4 (Jockusch and Schupp [9]; Igusa [7]; see also [5]). Are there minimal pairs in the (nonuniform or uniform) generic degrees?…”

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confidence: 99%