On 6/4/2020 4:07 AM, Bruno Marchal wrote:
On 2 Jun 2020, at 19:34, 'Brent Meeker' via Everything List
<everything-list@googlegroups.com> wrote:
On 6/2/2020 2:49 AM, Bruno Marchal wrote:
On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List
<everything-list@googlegroups.com> 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.
They bet on finite machines … including the universal machine, which I insist
is a finite machine. That is even the reason why I called it from times to
times universal number.
I recall that once we get the phi_i,
i = 1 to inf.
which can be defined in elementary arithmetic, we get all the universal
numbers, that is all u such that there phi_u(x, y) = phi_x(y), and such u can
be used to define all the recursive enumeration of all digital machines.
The implementation of this fine but universal machines are called (physical)
computer, and is the domain of expertise of IBM.
Bruno
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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