In <20120524215637.243$ZX@newsreader.com>, on 05/25/2012 at 01:56 AM, curt@kcwc.com (Curt Welch) said:
>You can't define existence by the convergence of an infinite process. >That's just absurd.
Graham Cooper doesn't understand the definition of the reals. However, one common definition is that a real is an equivalence class of Cauchy sequences in the rationals, and one theorem that applies to the reals however defined is that every Cauchy sequence converges. Note that a sequence is *not* a process, and there is no process involved.
>Some infinite processes converge _towards_ a thing that does exist.
Define process and convergence of processes.
>If you try to define rationals as what the processes converges to, >then we have the issue of 0.9999.. being a very different process >than 1.0000....
Only if you have a really bizarre definition of convergence. IAC, it has nothing to do with convergence of sequences, which is what is relevant.
>So now we have an infinite number of processes producing >different sequences, which must be claimed to be "the same real".
The limit of a sequence is not the sequence.
>What a disaster.
If a_1 = 1 for all i, b_1 = 0 and b_i = 1 for all i>0, does it bother you that a and b both converge to 1? If so, why?
>The diagonal "proof" is no longer a proof if we do this for example
Wrong again, because it does not involve processes at all.
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