Separating Fragments of Wlem, Lpo, and Mp

Hendtlass, Matthew; Lubarsky, Robert S.

doi:10.1017/jsl.2016.38

Cited by 8 publications

(5 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Together we have the desired contradiction to (7.2). This nicely fits in with a result of Lubarsky and Hendtlass who showed in [67] that there is a topological model satisfying LLPO, but not LPO, which means that that model also does not satisfy WMP.…”

Section: Reverse Reverse Mathematicssupporting

confidence: 90%

Constructive Reverse Mathematics

Diener¹

2018

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: Reverse Reverse Mathematicssupporting

confidence: 90%

Constructive Reverse Mathematics

Diener¹

2018

Preprint

View full text Add to dashboard Cite

show abstract

“…We note that ACC N ≤ sW ACC n+1 ≤ sW ACC n holds for all n ≥ 2 and that the condition ACC n+1 < W ACC n for all n ≥ 2 implies the strictness of the reduction ACC N < W ACC n+1 . We note that the LLPO n hierarchy has recently also been separated over IZF+DC [28].…”

Section: Choice and Omnisciencementioning

confidence: 86%

On the Uniform Computational Content of Computability Theory

2017

View full text Add to dashboard Cite

We demonstrate that the Weihrauch lattice can be used to classify the uniform computational content of computability-theoretic properties as well as the computational content of theorems in one common setting. The properties that we study include diagonal non-computability, hyperimmunity, complete consistent extensions of Peano arithmetic, 1-genericity, Martin-Löf randomness, and cohesiveness. The theorems that we include in our case study are the low basis theorem of Jockusch and Soare, the Kleene-Post theorem, and Friedberg's jump inversion theorem. It turns out that all the aforementioned properties and many theorems in computability theory, including all theorems that claim the existence of some Turing degree, have very little uniform computational content: they are located outside of the upper cone of binary choice (also known as LLPO); we call problems with this property indiscriminative. Since practically all theorems from classical analysis whose computational content has been classified are discriminative, our observation could yield an explanation for why theorems and results in computability theory typically have very few direct consequences in other disciplines such as analysis. A notable exception in our case study is the low basis theorem which is discriminative. This is perhaps why it is considered to be one of the most applicable theorems in computability theory. In some cases a bridge between the indiscriminative world and the discriminative world of classical mathematics can be established via a suitable residual operation and we demonstrate this in the case of the cohesiveness problem and the problem of consistent complete extensions of Peano arithmetic. Both turn out to be the quotient of two discriminative problems.

show abstract

“…Hendtlass and Lubarsky showed in [9] that LLPO n+1 is independent of LLPO n over IZF + DC using topological models. We obtain here a similar separation result.…”

Section: Consistency Of Church's Thesis With Llpo Nmentioning

confidence: 99%

“…The topologies L n correspond to certain variants of the lesser limited principle of omniscience, LLPO, which were first studied by Richman in [20], and are denoted LLPO n . We will use these models to give a new proof of a theorem due to Hendtlass and Lubarsky in [9]: LLPO n+1 is strictly weaker than LLPO n . This answers positively a question raised by Hendtlass: is there a variant of Lifschitz realizability that separates LLPO n from LLPO n+1 ?…”

Section: Introductionmentioning

confidence: 99%

Lifschitz Realizability as a Topological Construction

Rathjen¹,

Swan²

2020

J. symb. log.

View full text Add to dashboard Cite

We develop a number of variants of Lifschitz realizability for $\mathbf {CZF}$ by building topological models internally in certain realizability models. We use this to show some interesting metamathematical results about constructive set theory with variants of the lesser limited principle of omniscience including consistency with unique Church’s thesis, consistency with some Brouwerian principles and variants of the numerical existence property.

show abstract

Separating Fragments of Wlem, Lpo, and Mp

Cited by 8 publications

References 26 publications

Constructive Reverse Mathematics

Constructive Reverse Mathematics

On the Uniform Computational Content of Computability Theory

Lifschitz Realizability as a Topological Construction

Contact Info

Product

Resources

About