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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: 385
Registered: 7/12/10
Foundations of real numbers
Posted: Oct 10, 2012 1:30 PM
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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
Read Foundations of real numbers
Paul
10/10/12
Read Re: Foundations of real numbers
Ken.Pledger@vuw.ac.nz
10/13/12
Read Re: Foundations of real numbers
Graham Cooper
10/13/12
Read Re: Foundations of real numbers
Frederick Williams
10/13/12
Read Re: Foundations of real numbers
Graham Cooper
10/11/12
Read Re: Foundations of real numbers
Peter Webb
10/11/12
Read Re: Foundations of real numbers
Shmuel (Seymour J.) Metz
10/11/12
Read Re: Foundations of real numbers
Michael Stemper
10/12/12
Read Re: Foundations of real numbers
Peter Webb
10/12/12
Read Re: Foundations of real numbers
Virgil
10/12/12
Read Re: Foundations of real numbers
LudovicoVan
10/12/12
Read Re: Foundations of real numbers
Michael Stemper
10/12/12
Read Re: Foundations of real numbers
Peter Webb
10/13/12
Read Re: Foundations of real numbers
Shmuel (Seymour J.) Metz
10/12/12
Read Re: Foundations of real numbers
Shmuel (Seymour J.) Metz
10/12/12
Read Re: Foundations of real numbers
Peter Webb
10/13/12
Read Re: Foundations of real numbers
Peter Webb
10/13/12
Read Re: Foundations of real numbers
Frederick Williams
10/11/12
Read Re: Foundations of real numbers
Robin Chapman
10/11/12
Read Re: Foundations of real numbers
Frederick Williams
10/11/12
Read Re: Foundations of real numbers
Frederick Williams
10/11/12
Read Re: Foundations of real numbers
Herman Rubin
10/11/12
Read Re: Foundations of real numbers
Michael Stemper
10/11/12
Read Re: Foundations of real numbers
Brian Q. Hutchings
10/12/12
Read Re: Foundations of real numbers
G. A. Edgar
10/14/12
Read Re: Foundations of real numbers
gnasher729
10/14/12
Read Re: Foundations of real numbers
Graham Cooper

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