- #71
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
- 14,983
- 28
I can go through the proof that a convergent countable sequence of hyperreals must be eventually constant from the ultrapdroduct construction, if you like. It's an ugly mess, though!
I don't think it really matters though -- I get the suspicion you're so busy trying to refute "*R is complete" that there's a subtle difference between that and what I'm saying! (And as I mentioned, it took me a while to pin down this subtlety myself!)
You are correct in saying *R is not (externally) Dedekind complete.
But I'm saying *R is internally Dedekind complete.
You are correct in saying the N-indexed sequence {1 - 10^(-n)} does not converge to 1 in *R. (it doesn't converge at all).
But I'm saying the *N-indexed sequence {1 - 10^(-n)} does converge to 1.
Also, I'm saying that the correct notion of "decimal" when looking at the hyperreal numbers involves having *N-many decimal places (not N-many).
I don't think it really matters though -- I get the suspicion you're so busy trying to refute "*R is complete" that there's a subtle difference between that and what I'm saying! (And as I mentioned, it took me a while to pin down this subtlety myself!)
You are correct in saying *R is not (externally) Dedekind complete.
But I'm saying *R is internally Dedekind complete.
You are correct in saying the N-indexed sequence {1 - 10^(-n)} does not converge to 1 in *R. (it doesn't converge at all).
But I'm saying the *N-indexed sequence {1 - 10^(-n)} does converge to 1.
Also, I'm saying that the correct notion of "decimal" when looking at the hyperreal numbers involves having *N-many decimal places (not N-many).
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