Cousin’s lemma in second-order arithmetic

Barrett, Jordan Mitchell; Downey, Rodney G.; Greenberg, Noam

doi:10.1090/bproc/111

Cited by 3 publications

(21 citation statements)

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“…For the third item, to find a point First of all, after introducing the Baire classes, Baire notes that Baire 2 functions can be represented by repeated limits as in (5) (see [2, p. 69]). Given ∃ 2 , effectively Baire 2 functions are essentially the representation of Baire 2 functions used in second-order arithmetic ( [3]).…”

Section: Theoremmentioning

confidence: 99%

The non-normal abyss in Kleene's computability theory

Sanders¹

2023

Preprint

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Kleene's computability theory based on his S1-S9 computation schemes constitutes a model for computing with objects of any finite type and extends Turing's 'machine model' which formalises computing with real numbers. A fundamental distinction in Kleene's framework is between normal and non-normal functionals where the former compute the associated Kleene quantifier ∃ n and the latter do not. Historically, the focus was on normal functionals, but recently new non-normal functionals have been studied, based on well-known theorems like the uncountability of the reals. These new non-normal functionals are fundamentally different from historical examples like Tait's fan functional: the latter is computable from ∃ 2 while the former are only computable in ∃ 3 . While there is a great divide separating ∃ 2 and ∃ 3 , we identify certain closely related non-normal functionals that fall on different sides of this abyss. Our examples are based on mainstream mathematical notions, like quasi-continuity, Baire classes, and semi-continuity.⋆ This research was supported by the Deutsche Forschungsgemeinschaft (DFG) (grant nr. SA3418/1-1) and the Klaus Tschira Boost Fund (grant nr. GSO/KT 43).Recently, Dag Normann and the author have identified new non-normal functionals based on mainstream theorems like e.g. the Heine-Borel theorem, the Jordan decomposition theorem, and the uncountability of R ([19-21, 23-25]). These non-normal functionals are very different as follows: Tait's fan functional is computable in ∃ 2 , making it rather tame; by contrast the following non-normal operation is not computable in any S 2 k , where the latter decides Π 1 k -formulas.Clearly, this operation witnesses the basic fact there is no injection from the unit interval to the naturals. The operation in (1) can be performed by ∃ 3 , which follows from some of the many proofs that R is uncountable. Essentially all the non-normal functionals studied in [19][20][21][23][24][25] compute the operation in (1), or some equally hard variation.

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Section: Theoremmentioning

confidence: 99%

The non-normal abyss in Kleene's computability theory

Sanders¹

2023

Preprint

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“…We say that f is effectively Baire 0 if it is computable, and effectively Baire n + 1, if it is the pointwise limit of a computable collection of effectively Baire n functions. In [6] it is shown that if α is a computable ordinal, then there is an effective Baire 2 function f such that any -fine partition computes H (α) .…”

Section: Corollary 43mentioning

confidence: 99%

“…which is (uniformly) continuous if f was. 6 Since our language is merely td u, the atomic formulae are just d (x, y) and the constant functions 0 and 1. We close these formulae under finite iterations of sup, inf, ^, _, Definition 3.6.…”

Section: Continuous Logic and Decidabilitymentioning

confidence: 99%

“…Another more recent application of Π 0 1 classes in computable analysis is by Barrett, Downey, and Greenberg [6], and concerns Cousin's lemma. This is a core lemma in the theory of the Denjoy integral.…”

Section: Corollary 43mentioning

confidence: 99%

“…Using similar codings to those of Theorem 4.2, we have the following theorem. Theorem 4.6 (Barrett, Downey, and Greenberg [6]). There is a computable gauge with no computable -fine partition.…”

Section: Corollary 43mentioning

confidence: 99%

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Computably Compact Metric Spaces

DOWNEY¹,

Melnikov²

2023

Bull. symb. log

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We give a systematic technical exposition of the foundations of the theory of computably compact metric spaces. We discover several new characterizations of computable compactness and apply these characterizations to prove new results in computable analysis and effective topology. We also apply the technique of computable compactness to give new and less combinatorially involved proofs of known results from the literature. Some of these results do not have computable compactness or compact spaces in their statements, and thus these applications are not necessarily direct or expected.

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The Biggest Five of Reverse Mathematics

Normann,

Sanders

2023

J. Math. Log.

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The aim of Reverse Mathematics (RM for short) is to find the minimal axioms needed to prove a given theorem of ordinary mathematics. These minimal axioms are almost always equivalent to the theorem, working over the base theory of RM, a weak system of computable mathematics. The Big Five phenomenon of RM is the observation that a large number of theorems from ordinary mathematics are either provable in the base theory or equivalent to one of only four systems; these five systems together are called the ‘Big Five’. The aim of this paper is to greatly extend the Big Five phenomenon as follows: there are two supposedly fundamentally different approaches to RM where the main difference is whether the language is restricted to second-order objects or if one allows third-order objects. In this paper, we unite these two strands of RM by establishing numerous equivalences involving the second-order Big Five systems on one hand, and well-known third-order theorems from analysis about (possibly) discontinuous functions on the other hand. We both study relatively tame notions, like cadlag or Baire 1, and potentially wild ones, like quasi-continuity. We also show that slight generalizations and variations of the aforementioned third-order theorems fall far outside of the Big Five.

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Cousin’s lemma in second-order arithmetic

Cited by 3 publications

References 11 publications

The non-normal abyss in Kleene's computability theory

The non-normal abyss in Kleene's computability theory

Computably Compact Metric Spaces

The Biggest Five of Reverse Mathematics

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