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Pseudo-ordered fields

Published online by Cambridge University Press:  03 November 2016

Peter Csontos ,
Christopher L. Morgan  and
Kenneth Rebman
Peter Csontos
Affiliation:
California State University, Hayward, California 94542, USA
Christopher L. Morgan
Affiliation:
California State University, Hayward, California 94542, USA
Kenneth Rebman
Affiliation:
California State University, Hayward, California 94542, USA
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Extract

A danger in teaching familiar material is oversimplification. In this article, we shall explore an example of this.

One of the authors observed a colleague (at another institution) introduce his students to the ordering of the real numbers by stating that < is a relation on ℝ which satisfies the following axiom system:

“For all x, y, z in ℝ:

  1. 1. Exactly one of x = y, x < y, y < x holds.

  2. 2. If x < y and 0 < z, then xz < yz.

  3. 3. If x < y, then x + z < y + z.”

Type
Research Article
Information
The Mathematical Gazette , Volume 58 , Issue 405 , October 1974 , pp. 195 - 198
Copyright
Copyright © Mathematical Association 1974

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References

1. Niven, I. and Zuckerman, H. S., An introduction to the theory of numbers (2nd edn.). Wiley (New York, 1966).Google Scholar
2. Dickson, L. E., Linear groups (with an exposition of the Galois field theory). Dover (1958).Google Scholar