Another Unique Weak König’s Lemma WKL!!

Moschovakis, Joan Rand

doi:10.1515/9783110324921.343

“…Notice that our notion of WKL!!! differs only slightly from the one of J. Moschovakis, who first considered it [97]. In her version, named WKL!…”

Section: (Wkl!!!)mentioning

confidence: 88%

Constructive Reverse Mathematics

Diener¹

2018

Preprint

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However, there are also uses of ACC and ADC of a more structural kind. The proof that LLPO implies WKL needs ACC not because we do not assume that we have a nice representations of binary trees, but because we need to use LLPO countable many times.To gain more insights into phenomena of the second kind, attempts have been made to simply move choice principles into the list of principle studied [17]. And indeed, the original plan for this thesis was to work choice-sensitive and distinguish, for example, between the sequential version of LPO and the real version. However this quickly turned out to be too ambitious a project. The big picture (Section 6.5) is already very complicated and the number of principles would multiply in the absence of choice. Any attempt to do CRM without the use of ACC or ADC would need to find a way to present results in a way that highlights the interesting issues of the second kind and somehow manages to not give too much prominence to issues of the first kind. Overview and PlanContrary to Simpson style reverse mathematics, in which most theorems fall into one of the "big five" categories, 8 there is a plethora of principles that have been considered in constructive reverse mathematics, with a quick count totalling about 17 major ones. We believe that the presentation we will give is a sensible way to group them. If we consider the big three varieties CLASS, INT, RUSS (see Section 7.1) there are seven possible combinations of these varieties such that a principle is true in at least one of them and possibly fails to hold in others. Five of these combinations form our first five chapters.Chapter 1: Omniscience principles which are true classically, but not in INT or RUSS. CLASS RUSS INTChapter 2: Markov's principle and its weakenings which are true in CLASS and RUSS. Actually, WMP is true everywhere, but fits better into this chapter than into the chapter about BD-N.

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“…In [7], equivalents are given for what is there called FAN and WKL, although their FAN is actually D-FAN, and it is at least asked how much stronger WKL is than FAN. The one proof we have been able to find of some fan theorem not implying WKL is in [19], where once again the fan principle used is D-FAN. For what it's worth, that argument, like ours, uses relative realizability [8], albeit with K 2 realizability.…”

Section: Fan and Wklmentioning

confidence: 99%

Realizability Models Separating Various Fan Theorems

Lubarsky

¹

,

Rathjen

²

2013

Lecture Notes in Computer Science

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“…In this paper, we decompose weak König’s lemma with a uniqueness hypothesis from Moschovakis [3], abbreviated WKL!!, into logical and function–existence principles in a recent framework of CRM.…”

Section: Introductionmentioning

confidence: 99%

“…More recently, Moschovakis [3] introduced another version of unique weak König’s lemma WKL!! where the uniqueness hypothesis is defined in a naive way: any two paths through a given infinite binary tree are the same (see (2.3) for the precise definition). Then she showed that WKL!! can be decomposed into the Π10-fragment of De Morgan’s law (under the name of normalMP∨ from Ishihara [12]) and the double negated variant of WKL (see [3, Section 4]), and also that WKL!! lies strictly between WKL! and WKL (see [3, Section 5]).…”

Section: Introductionmentioning

confidence: 99%

“…More recently, Moschovakis [3] introduced another version of unique weak König’s lemma WKL!! where the uniqueness hypothesis is defined in a naive way: any two paths through a given infinite binary tree are the same (see (2.3) for the precise definition). Then she showed that WKL!! can be decomposed into the Π10-fragment of De Morgan’s law (under the name of normalMP∨ from Ishihara [12]) and the double negated variant of WKL (see [3, Section 4]), and also that WKL!! lies strictly between WKL! and WKL (see [3, Section 5]). However, the decomposition result is not satisfactory for the following reasons: The base theory M of the decomposition result is based on intuitionistic logic but contains the unique choice principle from numbers to numbers normalAC0,0! as an axiom scheme.…”

Section: Introductionmentioning

confidence: 99%

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On the decomposition of WKL!!

Fujiwara

¹

,

Nemoto

²

2023

Phil. Trans. R. Soc. A.

1

0

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From the conceptual viewpoint, many mathematical propositions implicitly contain at least two kinds of principle. One is a logical principle such as the law-of-excluded-middle or De Morgan’s law. Another is a function–existence principle. For both conceptual and practical reasons, it is an interesting enterprise to calibrate how amount of logical and function–existence principles are implicit in mathematical theorems and axioms. This is the topic of constructive reverse mathematics, which specifies necessary and sufficient axioms to prove each mathematical proposition constructively. In this paper, we decompose weak König’s lemma with a uniqueness hypothesis WKL ! ! by Moschovakis, into logical and function–existence principles in a recent framework of constructive reverse mathematics. This article is part of the theme issue ‘Modern perspectives in Proof Theory’.

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Another Unique Weak König’s Lemma WKL!!

Cited by 8 publications

References 6 publications

Constructive Reverse Mathematics

Constructive Reverse Mathematics

Realizability Models Separating Various Fan Theorems

On the decomposition of WKL!!

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