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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


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.



