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Validating Brouwer's continuity principle for numbers using named exceptions

VINCENT RAHLI, MARK BICKFORD
2017 Mathematical Structures in Computer Science  
Using these new features, we prove a version of Brouwer's continuity principle for numbers. We also provide a simpler proof of a weaker version of this principle that only uses diverging terms.  ...  Using continuity and the fan theorem, we prove important results of Intuitionistic Mathematics: Brouwer's continuity theorem, bar induction on monotone bars and the negation of the law of excluded middle  ...  Constable, Rich Eaton, Evan Moran, and Ross Tate for their helpful criticism.  ... 
doi:10.1017/s0960129517000172 fatcat:n7xzyn2zh5cgji26zrj6lrykw4

Book Review: The foundations of intuitionistic mathematics

Errett Bishop
1965 Bulletin of the American Mathematical Society  
Brouwer's principle and the Bar Theorem are used to show that every real-valued function on the interval [0, l] is uniformly continuous, a famous result of Brouwer.  ...  Brouwer's principle roughly stated says that if ƒ is any function from the set of natural number sequences to the natural numbers then to every natural number sequence {a n }=a corresponds an integer N  ... 
doi:10.1090/s0002-9904-1965-11412-4 fatcat:f7jd5epnc5g7zbimutcpv5ldbq

Exercising Nuprl's Open-Endedness [chapter]

Vincent Rahli
2016 Lecture Notes in Computer Science  
versions of the axiom of choice and of Brouwer's bar induction principle.  ...  We have recently exercised Nuprl's open-endedness by adding nominal features to Nuprl in order to prove a version of Brouwer's continuity principle, as well as choice sequences in order to prove truncated  ...  For that, following Longley's method [31] , we used named exceptions as a probing mechanism to compute the modulus of continuity of a function. (3) We have proved the validity of versions of Brouwer's  ... 
doi:10.1007/978-3-319-42432-3_3 fatcat:exbfyiolojditotrvifbhkprva

Book Review: L. E. J. Brouwer collected works, Volume I, Philosophy and foundations of mathematics

G. Kreisel
1977 Bulletin of the American Mathematical Society  
(Quite generally, Brouwer's insistence on r.n.g. is appropriate when continuity is paramount: a continuous mapping from R into a discrete space is constant, but not for r.n.g.)  ...  (iii) In analysis, one asks about Brouwer's fixed point theorem, for the uniform convergence topology of mappings ƒ of, say S 2 H» S 2 : IS there a continuous J: ƒ H> X E S 2 such that ƒ[?(ƒ)] = ?  ...  Hilbert wanted to 'justify' the latter by using (i) formalizations <5 of valid principles for then-current mathematical concepts and (ii) proving the consistency of 5"(tacitly, constructively): he was  ... 
doi:10.1090/s0002-9904-1977-14185-2 fatcat:ax5t5iepyncidgj6fk4wf6r4he

At the Heart of Analysis: Intuitionism and Philosophy

Charles McCarty
2006 Philosophia Scientiæ  
Acknowledgements This paper is dedicated to Dirk van Dalen in gratitude for years of inspiration and encouragement.  ...  For example, Brouwer's Theorem itself entails the failure of TND. Were TND generally valid, every real number would be rational or irrational.  ...  In consequence, the intuitionist feels herself under no more obligation to accept the validity of TND than she is to accept the validity of other plainly invalid principles, for example, p ∨ ¬q.  ... 
doi:10.4000/philosophiascientiae.411 fatcat:bwpcdtglabeczfmqhpty4gwbpq

Luitzen Egbertus Jan Brouwer. 1881-1966

G. Kreisel, M. H. A. Newman
1969 Biographical Memoirs of Fellows of the Royal Society  
have a constructiv consistency proof for the formal principles of analysis, and this would be a minimum requirement for a positive answer, since those principles are evidently valid for the (non-constructive  ...  For sequences s taking a bounded number of values, p is uniformly con tinuous (pointwise continuity was assumed from the start).  ...  BIBLIOGRAPHY Titles of Dutch publications are followed, in brackets, by a free translation in modem terms, for example, in (112) Brouwer's 'punt' is translated by 'point generator' and his 'kern', which  ... 
doi:10.1098/rsbm.1969.0002 fatcat:hzk643xinbazxdjggcg24ykea4

Page 26 of Mathematical Reviews Vol. , Issue 83a [page]

1983 Mathematical Reviews  
Math. 24 (1978), no. 5, 427-436; MR _ 80b:03096] to obtain a model of intuitionistic set theory satisfying the Kripke schema, Brouwer’s principle for numbers, Brouwer’s principle for functions with the  ...  Another model satisfies Kripke’s schema and Brouwer’s principle for numbers without parameters but not bar induction. The author stresses the topos-theoretic point of view.  ... 

Bar induction: The good, the bad, and the ugly

Vincent Rahli, Mark Bickford, Robert L. Constable
2017 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)  
We present an extension of the computation system and logic of the Nuprl proof assistant with intuitionistic principles, namely versions of Brouwer's bar induction principle, which is equivalent to transfinite  ...  We write B k for N N k , where k is a natural number and N k is the type of natural numbers strictly less than k. We use Π and Σ in lieu of the constructive logical quantifiers ∀ and ∃, respectively.  ...  Acknowledgements We thank David Guaspari and Evan Moran for their helpful criticism.  ... 
doi:10.1109/lics.2017.8005074 dblp:conf/lics/RahliBC17 fatcat:jilht2aeozh6zor7bhcqgx32se

Brouwer's Incomplete Objects

Joop Niekus
2010 History and Philosophy of Logic  
The theory of the idealized mathematician has been developed to formalize a method that is characteristic for Brouwer's papers after 1945.  ...  We do not use an idealized mathematician. We claim that it is the systematic application of incomplete sequences, already introduced by Brouwer in 1918, that makes the method special.  ...  I thank the organization of the LIO for accepting the project. I thank Olivier Roy for his translation of the abstract. I am much indebted to  ... 
doi:10.1080/01445340903445071 fatcat:7zixwklkcrbmrd2st5k4p5jp5q

Intuition, iteration, induction [article]

Mark van Atten
2015 arXiv   pre-print
In Mathematical Thought and Its Objects, Charles Parsons argues that our knowledge of the iterability of functions on the natural numbers and of the validity of complete induction is not intuitive knowledge  ...  But I will try to make two points: (1) Using elements from Husserl and from Brouwer, Brouwer's claims can be justified in more detail than he has done; (2) There are certain elements in Parsons' discussion  ...  , Charles Parsons argues that our knowledge that these two general principles are valid is not intuitive knowledge.  ... 
arXiv:1510.01094v1 fatcat:l7y4skiptzazbce6tmhkfnfb2i

The Creating Subject, the Brouwer–Kripke Schema, and infinite proofs

Mark van Atten
2018 Indagationes mathematicae  
Abstract Kripke's Schema (better the Brouwer-Kripke Schema) and the Kreisel-Troelstra Theory of the Creating Subject were introduced around the same time for the same purpose, that of analysing Brouwer's  ...  Then I discuss the original use of the Schema and the Theory, their justification from a Brouwerian perspective, and instances of the Schema that can in fact be found in Brouwer's own writings.  ...  WC-N can be strengthened to Principle 25 (Continuity for Numbers).  ... 
doi:10.1016/j.indag.2018.06.005 fatcat:4qgl4bi5trd6leisflx5hoyy4q

Intuitionistic Mathematics and Logic [article]

Joan R. Moschovakis, Garyfallia Vafeiadou
2020 arXiv   pre-print
theories, and sketch his use of g\"odel numbers of recursive functions to realize sentences of intuitionistic arithmetic including a form of Church's Thesis.  ...  for logic and mathematics.  ...  Implicit in the use of weak counterexamples is Brouwer's continuity principle, to be explained below. The continuum cannot be ordered.  ... 
arXiv:2003.01935v1 fatcat:ipbk2fmjxnd6zasjpcijdcqoxq

The Creating Subject, the Brouwer-Kripke Schema, and infinite proofs [article]

Mark van Atten
2018 arXiv   pre-print
Kripke's Schema (better the Brouwer-Kripke Schema) and the Kreisel-Troelstra Theory of the Creating Subject were introduced around the same time for the same purpose, that of analysing Brouwer's 'Creating  ...  Then I discuss the original use of the Schema and the Theory, their justification from a Brouwerian perspective, and instances of the Schema that can in fact be found in Brouwer's own writings.  ...  WC-N can be strengthened to Principle 25 (Continuity for Numbers).  ... 
arXiv:1805.00404v1 fatcat:mihkswwo5bhnjidf4mwdechwni

Arguments for the Continuity Principle

Mark van Atten, Dirk van Dalen
2002 Bulletin of Symbolic Logic  
One, the principle of bar induction, will not concern us here. The other, the continuity principle for numbers, occurs for the first time in print in [4].  ...  Let us note first that in one particular case the principle is obvious indeed, namely in the case of the lawless sequences.  ...  We wish to thank Andreas Blass, Michael Dummett, Hajime Ishihara, Anne Troelstra, Albert Visser and the referee for their comments, which led to several improvements.  ... 
doi:10.2178/bsl/1182353892 fatcat:7yrrivv62vb3zezvcmsyseumz4

A nominal exploration of intuitionism

Vincent Rahli, Mark Bickford
2016 Proceedings of the 5th ACM SIGPLAN Conference on Certified Programs and Proofs - CPP 2016  
Using these new features, we prove a version of Brouwer's Continuity Principle for numbers. We also provide a simpler proof of a weaker version of this principle that only uses diverging terms.  ...  Using continuity and the fan theorem we prove important results of Intuitionistic Mathematics: Brouwer's continuity theorem and bar induction on monotone bars.  ...  Constable, Ross Tate, Rich Eaton, Abhishek Anand, and Evan Moran for their helpful criticism.  ... 
doi:10.1145/2854065.2854077 dblp:conf/cpp/RahliB16 fatcat:55xfsdlzcrc3zoxyhv64kqrz3a
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