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Topic: Foundations of real numbers
Replies: 26   Last Post: Oct 14, 2012 7:35 PM

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 Paul Posts: 780 Registered: 7/12/10
Foundations of real numbers
Posted: Oct 10, 2012 1:30 PM

It seems ridiculous to me to define the real numbers by using the least-upper-bound property as an axiom, although this seems to be the most common way.
The definition of the reals should formalise the way we approximate pi by a decimal expansion: 3, 3.1, 3.14 etc.

So the Cauchy sequence definition is much better, and I like the Dedekind cuts definition too. I don't at all understand how it makes sense to regard the least-upper-bound property of the reals as an axiom. The least-upper-bound property should be regarded as a theorem, not an axiom.

Why did the least-upper-bound-property-as-an-axiom approach become so prevalent? If you define real numbers that way, the correspondence between our intuitive sense of real numbers and the formalisation is so much less clear than with either Dedekind cuts or Cauchy sequences.

Paul Epstein

Date Subject Author
10/10/12 Paul
10/10/12 Ken.Pledger@vuw.ac.nz
10/13/12 Graham Cooper
10/13/12 Frederick Williams
10/13/12 Graham Cooper
10/11/12 Peter Webb
10/11/12 Shmuel (Seymour J.) Metz
10/11/12 Michael Stemper
10/12/12 Peter Webb
10/12/12 Virgil
10/12/12 LudovicoVan
10/12/12 Michael Stemper
10/12/12 Peter Webb
10/13/12 Shmuel (Seymour J.) Metz
10/12/12 Shmuel (Seymour J.) Metz
10/12/12 Peter Webb
10/13/12 Peter Webb
10/13/12 Frederick Williams
10/11/12 Robin Chapman
10/11/12 Frederick Williams
10/11/12 Frederick Williams
10/11/12 Herman Rubin
10/11/12 Michael Stemper
10/11/12 Brian Q. Hutchings
10/12/12 G. A. Edgar
10/14/12 gnasher729
10/14/12 Graham Cooper