On 6/2/2020 2:49 AM, Bruno Marchal wrote:
On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List
<[email protected]> wrote:
On 6/1/2020 2:08 AM, Bruno Marchal wrote:
Brent suggest that we might recover completeness by restricting N to a finite
domain. That is correct, because all finite function are computable, but then,
we have incompleteness directly with respect to the computable functions, even
limited on finite but arbitrary domain. In fact, that moves makes the computer
simply vanishing, and it makes Mechanism not even definable or expressible.
That's going to come as a big shock to IBM stockholders.
Why? On the contrary. IBM bets on universal machine
No, they bet only on finite machines, and they will be very surprised to
hear that they have vanished.
Brent
and know well what is a computer: a finite arithmetical being in touch with the
infinite, and indeed, always asking for more memory, which is the typical
symptom of liberty/universality. IBM might be finitist, like Mechanism, but is
not ultrafinist at all. Anyway, mathematically, Mechanism is consistent with
ulrafinitsim, even if to prove this, you need to go beyond finitism, (but then
that’s the case for all consistent theory: none can prove its own consistency
once “rich enough” (= just Turing universal, not “Löbian”).
Bruno
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