# Arguments for the continuity principle

@article{Atten2002ArgumentsFT, title={Arguments for the continuity principle}, author={Mark van Atten and Dirk van Dalen}, journal={Bull. Symb. Log.}, year={2002}, volume={8}, pages={329-347} }

There are two principles that lend Brouwer's mathematics the extra power beyond arithmetic. Both are presented in Brouwer's writings with little or no argument. 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]. It is formulated and immediately applied to show that the set of numerical choice sequences is not enumerable. In fact, the idea of the continuity property can be dated fairly…

## 41 Citations

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

- Philosophy2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2017

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 induction, and it is proved that these additions preserve N uprl's key metatheoretical properties such as consistency.

Validating Brouwer's continuity principle for numbers using named exceptions

- MathematicsMathematical Structures in Computer Science
- 2017

This paper extends the Nuprl proof assistant with named exceptions and handlers, as well as a nominal fresh operator, and proves a version of Brouwer's continuity principle for numbers and provides a simpler proof of this principle that only uses diverging terms.

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Of all the enigmas with which intuitionism confronted the world, that of “spread” was the most perplexing. Brouwer’s first definition [3], 13 lines long, was so exotic that hardly anyone could grasp…

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- Computer ScienceJ. ACM
- 2019

This work investigates the compatibility of several variants of BI with Constructive Type Theory (CTT), a dependent type theory in the spirit of Martin-Löf's extensional theory and provides novel insights regarding BI, such as the non-truncated version of BI on monotone bars being intuitionistically false.

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- 2016

This papers extends the Nuprl proof assistant with named exceptions and handlers, as well as a nominal fresh operator, to prove a version of Brouwer's Continuity Principle for numbers and provides a simpler proof of a weaker version of this principle that only uses diverging terms.

University of Birmingham Computability beyond Church-Turing via choice sequences

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- 2018

A new type theory BITT is developed, which is an extension of the type theory of the Nuprl proof assistant that embeds the notion of choice sequences and uses a Beth-like semantics to account for the dynamic nature ofchoice sequences.

The Oxford Handbook of Philosophy of Mathematics and Logic

- PhilosophyOxford handbooks in philosophy
- 2007

This volume covers these disciplines in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines, and is a ground-breaking reference like no other in its field.

Computability Beyond Church-Turing via Choice Sequences

- Computer ScienceLICS
- 2018

A new type theory BITT is developed, which is an extension of the type theory of the Nuprl proof assistant that embeds the notion of choice sequences and uses a Beth-like semantics to account for the dynamic nature of choice sequence.

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