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Topic: Proving archimedian property from "dedekind complete + orderer field"
Replies: 3   Last Post: Jul 28, 2013 12:55 PM

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David C. Ullrich

Posts: 3,238
Registered: 12/13/04
Re: Proving archimedian property from "dedekind complete + orderer field"
Posted: Jul 28, 2013 11:29 AM
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On Sun, 28 Jul 2013 03:16:24 -0700 (PDT), Lax Clarke
<lax.clarke@gmail.com> wrote:

>I'm confused as to why the archimedian property (unbounded natural numbers in reals) needs to be proven from {least upper bound property + ordered field}"
>
>Aren't the naturals "obviously" infinite (meaning provable without all the other machinery)? Because n+1 is bigger than n which are different natural numbers (you can prove n+1 > n > 0 using the order properties of reals).
>
>Is it my understanding that this proof is intended to exclude any case where the naturals could wrap around on each other like 1+1+1...+1=0 in a finite field?


Decided to answer this in a separate post so as not to disrupt the
flow of the other one.

No, that's really not it. "Ordered field" rules out 1 + ... + 1 = 0,
but "ordered field" is not enough to show that the naturals
are not bounded. There _are_ ordered fields in which the
naturals _are_ bounded. (Of course they cannot be
least-upper-bound complete, by the theorem we're
discussing.)

I suspect that people will mention the "nonstandard
reals", or "hyperreals" as an example. There are much
simpler examples.

For example, let F be the set of all rational functions:
quotients of (real) polynomials. Given a rational function
f, define f > 0 to mean that

there exists a real number x such that f(t) > 0
for all t > x.

It's not hard to show that F is now an ordered field.
Define f(t) = t. Then f is an uppper bound for the
naturals in F.

(The multiplicative identity in F is the constant
function 1, so the "naturals" are the elements of
the form 1 + ... + 1, that is, the constant functions
that take a natural number for their value.)

>Why can't we prove that from the totality properly of the order < ?
>
>How would one prove that the naturals inside the rationals (without the sup property which is used in the proof I talk about above) is unbounded above?
>
>Thanks for the help.





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