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6 - Arithmetic

from Part III - Where Are the Paradoxes?

Published online by Cambridge University Press:  08 October 2021

Zach Weber
Zach Weber
Affiliation:
University of Otago, New Zealand
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Summary

This chapter follows the axiomatic development of elementaryarithmetic. Starting from Peano’s postulates, somefundamental properties of natural numbers are derived, concerningaddition, multiplication, and order. If further assumptions aboutmathematical induction are made, then famous results about primenumbers can be established, including the fundamental theorem ofarithmetic (unique prime factorization) and Euclid’s theoremabout the infinitude of primes. Some simple facts about even and oddnumbers lead to the original limitive diagonal theorem: that thesquare root of 2 is irrational. Features of working withoutcontraction are highlighted.

Keywords

Peano arithmeticnatural numbersfundamental theorem of arithmeticirrationality of square root of 2
Type
Chapter
Information
Paradoxes and Inconsistent Mathematics , pp. 189 - 211
Publisher: Cambridge University Press
Print publication year: 2021

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  • Arithmetic
  • Zach Weber, University of Otago, New Zealand
  • Book: Paradoxes and Inconsistent Mathematics
  • Online publication: 08 October 2021
  • Chapter DOI: https://doi.org/10.1017/9781108993135.011
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  • Arithmetic
  • Zach Weber, University of Otago, New Zealand
  • Book: Paradoxes and Inconsistent Mathematics
  • Online publication: 08 October 2021
  • Chapter DOI: https://doi.org/10.1017/9781108993135.011
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Arithmetic
  • Zach Weber, University of Otago, New Zealand
  • Book: Paradoxes and Inconsistent Mathematics
  • Online publication: 08 October 2021
  • Chapter DOI: https://doi.org/10.1017/9781108993135.011
Available formats
×