SEPARATING FRAGMENTS OF WLEM, LPO, AND MP

@article{Hendtlass2016SEPARATINGFO,
  title={SEPARATING FRAGMENTS OF WLEM, LPO, AND MP},
  author={Matthew Hendtlass and Robert S. Lubarsky},
  journal={The Journal of Symbolic Logic},
  year={2016},
  volume={81},
  pages={1315 - 1343}
}
Abstract We separate many of the basic fragments of classical logic which are used in reverse constructive mathematics. A group of related Kripke and topological models is used to show that various fragments of the Weak Law of the Excluded Middle, the Limited Principle of Omniscience, and Markov’s Principle, including Weak Markov’s Principle, do not imply each other. 
View on Cambridge Press
math.fau.edu
12 Citations
Background Citations
3
Methods Citations
4
Results Citations
1

12 Citations

Lifschitz Realizability as a Topological Construction Michael Rathjen and

We develop a number of variants of Lifschitz realizability for CZF by building topological models internally in certain realizability models. We use this to show some interesting metamathematical…

Equivalents of disjunctive Markov's principle

The purpose of this short note is to point out a rich source of natural equivalents of the weak semi-intuitionistic principle MP$^\vee$ in reverse constructive mathematics: many simple theorems from…

Notions of Cauchyness and Metastability

It is shown that several weakenings of the Cauchy condition are all equivalent under the assumption of countable choice, and to what extent choice is necessary.

Point-Free Spectra of Linear Spreads

  • D. Wessel
  • Mathematics
    Mathesis Universalis, Computability and Proof
  • 2019
Scott’s multiple-conclusion entailment relations, originally conceived as a means to clarify several aspects of many-valued logic, and later put to great use in a revised Hilbert programme for…

Choice, extension, conservation. From transfinite to finite proof methods in abstract algebra

In this thesis several forms of the Kuratowski-Zorn Lemma are introduced and proved equivalent over constructive set theory, the notion of Jacobson radical is brought from commutative rings to a general ideal theory for single-conclusion entailment relations and a point-free version of Sikora’s theorem on spaces of orderings of groups is obtained.

Constructing the Constructible Universe Constructively

We study the properties of the constructible universe, L, over intuitionistic theories. We give an extended set of fundamental operations which is sufficient to generate the universe over…

Inner and Outer Models for Constructive Set Theories

We give a set-theoretic presentation of models of constructive set theory. The models are mostly Heyting-valued and Kripke models, and constructions that combine both of those ideas. The focus is on…

MARKOV’S PRINCIPLE, MARKOV’S RULE AND THE NOTION OF CONSTRUCTIVE PROOF

In [34] Georg Kreisel reflected at length on Church’s Thesis CT, the principle proposed in 1936 by Alonzo Church ([9]) as a definition: “We now define the notion . . . of an effectively calculable…

On the Uniform Computational Content of Computability Theory

It is demonstrated that the Weihrauch lattice can be used to classify the uniform computationalcontent of computability-theoretic properties as well as the computational content of theorems in one common setting and a bridge between the indiscriminative world and the discriminative World of classical mathematics can be established via a suitable residual operation.

Church's thesis and related axioms in Coq's type theory

The paper provides a partial answer to the question which axioms may preserve computational intuitions inherent to type theory, and which certainly do not, and is read as a broad survey ofAxioms in type theory.

References

SHOWING 1-10 OF 40 REFERENCES

An arithmetical hierarchy of the law of excluded middle and related principles

This paper compares, up to provability in intuitionistic arithmetic, subclassical principles like Markov's principle, (a function-free version of) weak Konig's lemma, Post's theorem, excluded middle for simply existential and simply universal statements, and many others.

A Calibration of Ineffective Theorems of Analysis in a Hierarchy of Semi-classical Logical Principles: (Extended Abstract)

A number of nonconstructive mathematical theorems are classified by means of a hierarchy of logical principles which are not included in intuitionistic logic by giving insights in both the scope of limit computable mathematics and its subsystems, and the nature of the theoreMS of classical mathematics considered.

Markov's principle, Church's thesis and Lindelöf's theorem

On Weak Markov's Principle

We show that the so-called weak Markov's principle (WMP) which states that every pseudo-positive real number is positive is underivable in ω ≔ E-HAω + AC. Since ω allows one to formalize (atl eastl…

Realizability Models Separating Various Fan Theorems

We develop a realizability model in which the realizers are the reals not just Turing computable in a fixed real but rather the reals in a countable ideal of Turing degrees. This is then applied to…

Weihrauch degrees, omniscience principles and weak computability

It is proved that parallelized LLPO is equivalent to Weak Kőnig's Lemma and hence to the Hahn–Banach Theorem in this new and very strong sense and any single-valued weakly computable operation is already computable in the ordinary sense.

Constructive Zermelo-Fraenkel set theory and the limited principle of omniscience

Calibration of Ineffective Theorems of Analysis in a Constructive Context

The main objective of this thesis is to investigate a hierarchy of logical principles not part of intuitionistic logic as developed by Akama et al. (2004). Intuitionistic logic is a basis for…

Realizing Brouwer's Sequences

Set theoretic foundations for constructive analysis

We introduce an extensional set theoretic formalism B, a subsystem of Zermelo set theory based on intuitionistic logic, which provides a set theoretic foundation for constructive analysis which is…