Infinite sets that Satisfy the Principle of Omniscience in any Variety of Constructive Mathematics

AbstractWe show that there are plenty of infinite sets that satisfy the omniscience principle, in a minimalistic setting for constructive mathematics that is compatible with classical mathematics. A first example of an omniscient set is the one-point compactification of the natural numbers, also known as the generic convergent sequence. We relate this to Grilliot's and Ishihara's Tricks. We generalize this example to many infinite subsets of the Cantor space. These subsets turn out to be ordinals in a constructive sense, with respect to the lexicographic order, satisfying both a well-foundedness condition with respect to decidable subsets, and transfinite induction restricted to decidable predicates. The use of simple types allows us to reach any ordinal below εQ, and richer type systems allow us to get higher.

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Constructive decidability of classical continuity

Mathematical Structures in Computer Science ◽

10.1017/s096012951300042x ◽

2014 ◽

Vol 25 (7) ◽

pp. 1578-1589

Author(s):

MARTÍN ESCARDÓ

Keyword(s):

Uniform Continuity ◽

Constructive Mathematics ◽

Natural Numbers ◽

Cantor Space ◽

Universal Quantifiers ◽

The One ◽

One Point Compactification ◽

Excluded Middle

We show that the following instance of the principle of excluded middle holds: any function on the one-point compactification of the natural numbers with values on the natural numbers is either classically continuous or classically discontinuous. The proof does not require choice and can be understood in any of the usual varieties of constructive mathematics. Classical (dis)continuity is a weakening of the notion of (dis)continuity, where the existential quantifiers are replaced by negated universal quantifiers. We also show that the classical continuity of all functions is equivalent to the negation of the weak limited principle of omniscience. We use this to relate uniform continuity and searchability of the Cantor space.

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Brouwer, L.E.J. (1881–1966)

Routledge Encyclopedia of Philosophy ◽

10.4324/9780415249126-dd3593-1 ◽

2018 ◽

Author(s):

Mark van Atten

Keyword(s):

Philosophy Of Mind ◽

Transcendental Phenomenology ◽

Full Professor ◽

Foundations Of Mathematics ◽

Constructive Mathematics ◽

Language Reform ◽

Classical Mathematics ◽

Mathematical Ontology ◽

The University ◽

Scale Alternative

L.E.J. Brouwer was a mathematician and philosopher. He graduated from the University of Amsterdam in 1907 and remained there, from 1913 to 1951, as full professor. Brouwer was a founding father of modern topology. In the foundations of mathematics he launched ‘intuitionism’: a mathematical ontology and epistemology, based on a philosophy of mind, that yields a form of constructive mathematics. Although intuitionism was designed as a Kantian approach, Brouwer’s conception of the intuition of time supports a much richer mathematics than Kant’s. Arguably, a closer affinity with Husserl’s transcendental phenomenology transpired as the latter was being developed. A by-product of intuitionism, intuitionistic logic, found application independently of the foundational programme. Intuitionism presented the first full-scale alternative to classical mathematics and logic. Brouwer was also interested in mysticism, and in language reform in the service of spiritual and political progress.

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A SOLUTION TO THE SURPRISE EXAM PARADOX IN CONSTRUCTIVE MATHEMATICS

The Review of Symbolic Logic ◽

10.1017/s1755020312000160 ◽

2012 ◽

Vol 5 (4) ◽

pp. 679-686 ◽

Cited By ~ 1

Author(s):

MOHAMMAD ARDESHIR ◽

RASOUL RAMEZANIAN

Keyword(s):

The Other ◽

Backward Induction ◽

Constructive Mathematics ◽

Constructive Logic ◽

Extensive Form ◽

Turing Computability ◽

Extensive Form Games ◽

Continuity Principle ◽

Classical Mathematics

AbstractWe represent the well-known surprise exam paradox in constructive and computable mathematics and offer solutions. One solution is based on Brouwer’s continuity principle in constructive mathematics, and the other involves type 2 Turing computability in classical mathematics. We also discuss the backward induction paradox for extensive form games in constructive logic.

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Against intuitionism: Constructive mathematics is part of classical mathematics

Journal of Philosophical Logic ◽

10.1007/bf00247188 ◽

1983 ◽

Vol 12 (2) ◽

Cited By ~ 9

Author(s):

W.W. Tait

Keyword(s):

Constructive Mathematics ◽

Classical Mathematics

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A finitary metalanguage for extended basic logic

Journal of Symbolic Logic ◽

10.2307/2267688 ◽

1952 ◽

Vol 17 (3) ◽

pp. 164-178 ◽

Cited By ~ 2

Author(s):

John Myhill

Keyword(s):

Combinatory Logic ◽

Transfinite Induction ◽

Basic Logic ◽

Elementary Number Theory ◽

Real Numbers ◽

Classical Analysis ◽

Theoretic Property ◽

Classical Mathematics ◽

Image Position ◽

Range Of Values

In a series of five papers, Fitch has constructed a system of combinatory logic K′ which is adequate for much of classical analysis, and is demonstrably consistent if we assume the validity of transfinite induction up to a certain ordinal (at present undetermined). The system has the following peculiarities:1.1. It is non-finitary, i.e. the class of Gödel-numbers of its theorems does not form the range of values of any recursive function (nor, as a matter of fact, of any function definable in elementary number theory).1.2. It does not permit quantification over real numbers; i.e. it contains no theorems of the form1.3. Because of 1.1, we cannot, except in trivial cases, construct actual proofs in K′; we have to resort to a metalanguage (of undetermined strength) in order to show that certain formulae are theorems of K′. Also because of 1.2, many important theorems of classical mathematics are not forthcoming in K′ itself, but only in the aforementioned metalanguage, e.g. in the formwhere ‘x’ is a syntactical rather than a numerical variable, a U-real is an expression of K′ rather than a number, and ‘ … x– – –’ ascribes a syntactical rather than a real number-theoretic property to x.The purpose of this paper is to construct a finitary metalanguage for K′ in which all of Fitch's important theorems may be proved.

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Theological Metaphors in Mathematics

Studies in Logic Grammar and Rhetoric ◽

10.1515/slgr-2016-0002 ◽

2016 ◽

Vol 44 (1) ◽

pp. 13-30 ◽

Cited By ~ 2

Author(s):

Stanisław Krajewski

Keyword(s):

Actual Infinity ◽

Mathematical Ideas ◽

Religious Background ◽

Human Power ◽

Classical Mathematics ◽

Theological Influences ◽

Infinite Sets ◽

Moscow School ◽

As If

Abstract Examples of possible theological influences upon the development of mathematics are indicated. The best known connection can be found in the realm of infinite sets treated by us as known or graspable, which constitutes a divine-like approach. Also the move to treat infinite processes as if they were one finished object that can be identified with its limits is routine in mathematicians, but refers to seemingly super-human power. For centuries this was seen as wrong and even today some philosophers, for example Brian Rotman, talk critically about “theological mathematics”. Theological metaphors, like “God’s view”, are used even by contemporary mathematicians. While rarely appearing in official texts they are rather easily invoked in “the kitchen of mathematics”. There exist theories developing without the assumption of actual infinity the tools of classical mathematics needed for applications (For instance, Mycielski’s approach). Conclusion: mathematics could have developed in another way. Finally, several specific examples of historical situations are mentioned where, according to some authors, direct theological input into mathematics appeared: the possibility of the ritual genesis of arithmetic and geometry, the importance of the Indian religious background for the emergence of zero, the genesis of the theories of Cantor and Brouwer, the role of Name-worshipping for the research of the Moscow school of topology. Neither these examples nor the previous illustrations of theological metaphors provide a certain proof that religion or theology was directly influencing the development of mathematical ideas. They do suggest, however, common points and connections that merit further exploration.

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Classical mathematics for a constructive world

Mathematical Structures in Computer Science ◽

10.1017/s0960129511000132 ◽

2011 ◽

Vol 21 (4) ◽

pp. 861-882 ◽

Cited By ~ 1

Author(s):

RUSSELL O'CONNOR

Keyword(s):

Classical Logic ◽

Type Theory ◽

Function Spaces ◽

Constructive Mathematics ◽

Constructive Logic ◽

Dependent Type Theory ◽

Interactive Theorem Provers ◽

Classical Function ◽

Classical Mathematics ◽

Dependent Type

Interactive theorem provers based on dependent type theory have the flexibility to support both constructive and classical reasoning. Constructive reasoning is supported natively by dependent type theory, and classical reasoning is typically supported by adding additional non-constructive axioms. However, there is another perspective that views constructive logic as an extension of classical logic. This paper will illustrate how classical reasoning can be supported in a practical manner inside dependent type theory without additional axioms. We will show several examples of how classical results can be applied to constructive mathematics. Finally, we will show how to extend this perspective from logic to mathematics by representing classical function spaces using a weak value monad.

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Synthesis of B-ring-fluorinated (−)-epicatechin gallate derivatives

Organic & Biomolecular Chemistry ◽

10.1039/d0ob00686f ◽

2020 ◽

Vol 18 (21) ◽

pp. 4024-4028

Author(s):

David D. S. Thieltges ◽

Kai D. Baumgarten ◽

Carina S. Michaelis ◽

Constantin Czekelius

Keyword(s):

Kinetic Resolution ◽

Convergent Sequence ◽

Epicatechin Gallate

Electronically modified, fluorinated catechins and epicatechins are enantioselectively synthesized in a short, convergent sequence via kinetic resolution.

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