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Lowness properties and K-triviality

André Nies

in Computability and Randomness

Published in print:
2009
Published Online:
May 2009
ISBN:
9780199230761
eISBN:
9780191710988
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199230761.003.0005
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter shows the equivalence of K-triviality and lowness for Martin–Löf randomness. This coincidence extends to other lowness properties, such as being a base for Martin–Löf randomness, and ... More


Martin-Löf randomness and its variants

André Nies

in Computability and Randomness

Published in print:
2009
Published Online:
May 2009
ISBN:
9780199230761
eISBN:
9780191710988
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199230761.003.0003
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter begins with a general background on how to introduce mathematical randomness notions. It then studies the central randomness notion of Martin–Löf, and gives some connections to ... More


Randomness and betting strategies

André Nies

in Computability and Randomness

Published in print:
2009
Published Online:
May 2009
ISBN:
9780199230761
eISBN:
9780191710988
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199230761.003.0007
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

Martingales form the mathematical counterpart of betting strategies. This chapter studies computable randomness, where the tests are computable martingales, and separates it from both Martin–Löf ... More


The Modern Theory of Induction

Kumaraswamy Velupillai

in Computable Economics: The Arne Ryde Memorial Lectures

Published in print:
2000
Published Online:
November 2003
ISBN:
9780198295273
eISBN:
9780191596988
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198295278.003.0005
Subject:
Economics and Finance, Macro- and Monetary Economics

A theory of induction, based on the formalism of algorithmic complexity theory, is introduced in this chapter. This is then used to underpin a theory of learning and a computable basis for inductive ... More


Modal Homotopy Type Theory: The Prospect of a New Logic for Philosophy

David Corfield

Published in print:
2020
Published Online:
March 2020
ISBN:
9780198853404
eISBN:
9780191888069
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198853404.001.0001
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

In[KF1] 1914, in an essay entitled ‘Logic as the Essence of Philosophy’, Bertrand Russell promised to revolutionize philosophy by introducing there the ‘new logic’ of Frege and Peano: “The old logic ... More


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