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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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Lax

Posts: 50
Registered: 3/18/07
Proving archimedian property from "dedekind complete + orderer field"
Posted: Jul 28, 2013 6:16 AM
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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?
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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