On Robust Theorems Due to Bolzano, Weierstrass, Jordan, and Cantor

Normann, Dag; Sanders, Sam

doi:10.1017/jsl.2022.71

Cited by 10 publications

(7 citation statements)

References 57 publications

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“…In conclusion, the operation (1) from Section 1.1 is computable in ∃ 3 but not in any S 2 k , as established in [22]. Many non-normal functionals exhibit the same 'computational hardness' and we merely view this as support for the development of a separate scale for classifying non-normal functionals.…”

Section: Some Comprehension Functionalsmentioning

confidence: 72%

“…covering theorems due to Heine-Borel, Vitali, and Lindelöf ( [20]), -the Baire category theorem ( [19]), -local-global principles like Pincherle's theorem ( [20]), -weak fragments of the Axiom of (countable) Choice ( [21]), -the Jordan decomposition theorem and related results ([23, 25]), -the uncountability of R ( [22,24]). Finally, the first example of a non-computable non-normal functional, Tait's fan functional ( [27]), is rather tame: it is computable in ∃ 2 .…”

Section: Preliminaries and Definitionsmentioning

confidence: 99%

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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: Some Comprehension Functionalsmentioning

confidence: 72%

Section: Preliminaries and Definitionsmentioning

confidence: 99%

The non-normal abyss in Kleene's computability theory

Sanders¹

2023

Preprint

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“…Now, arithmetical comprehension suffices to derive Theorem 1.4 for various kinds of second-order representations of BV -functions in [57,69]. By contrast, the results in [75][76][77][78] show that the Jordan decomposition theorem is even 'explosive': combining with the Suslin functional from Π 1 1 -CA ω 0 (see Section A.3), one derives Π 1 2 -CA 0 . Fourth, we shall make use of the following notion of (open and closed) set, which was studied in e.g.…”

Section: Definition 11 [Continuity]mentioning

confidence: 99%

“…Fourth, we shall make use of the following notion of (open and closed) set, which was studied in e.g. [75][76][77][78]89].…”

Section: Definition 11 [Continuity]mentioning

confidence: 99%

“…We assume basic familiarity with RM, including Kohlenbach's higher-order RM introduced in [55], while a brief sketch may be found in Section 1.3.1. Recent developments in higher-order RM, including our own, are in [70][71][72][73][74][75][76]. All equivalences are proved over Kohlenbach's base theory RCA ω 0 (or slight extensions), as defined in the appendix (Section A.1).…”

Section: Introductionmentioning

confidence: 99%

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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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Betwixt Turing and Kleene

Normann

Sanders

2021

Logical Foundations of Computer Science

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Turing's famous 'machine' model constitutes the first intuitively convincing framework for computing with real numbers. Kleene's computation schemes S1-S9 extend Turing's approach and provide a framework for computing with objects of any finite type. Various research programs have been proposed in which higher-order objects, like functions on the real numbers, are represented/coded as real numbers, so as to make them amenable to the Turing framework. It is then a natural question whether there is any significant difference between the Kleene approach or the Turing-approach-via-codes. Continuous functions being well-studied in this context, we study functions of bounded variation, which have at most countably many points of discontinuity. A central result is the Jordan decomposition theorem that a function of bounded variation on [0, 1] equals the difference of two monotone functions. We show that for this theorem and related results, the difference between the Kleene approach and the Turing-approach-via-codes is huge, in that full second-order arithmetic readily comes to the fore in Kleene's approach, in the guise of Kleene's quantifier ∃ 3 .

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On Robust Theorems Due to Bolzano, Weierstrass, Jordan, and Cantor

Cited by 10 publications

References 57 publications

The non-normal abyss in Kleene's computability theory

The non-normal abyss in Kleene's computability theory

The Biggest Five of Reverse Mathematics

Betwixt Turing and Kleene

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