On 6/5/2020 2:32 AM, Bruno Marchal wrote:
On 4 Jun 2020, at 20:28, 'Brent Meeker' via Everything List
<[email protected]> wrote:
On 6/4/2020 4:07 AM, Bruno Marchal wrote:
On 2 Jun 2020, at 19:34, 'Brent Meeker' via Everything List
<[email protected]> wrote:
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.
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.
That is the potential infinite,
No, you can't diagonalize on an infinity that is only potential.
that you already need for a concept like the square root of 2, used all the
time in elementary quantum mechanics.
And in every computer...which uses on finitely many bits.
Without it, neither CT, nor YD makes any sense.
CT doesn't. So much the worse for CT. YD makes better sense since the
doctor can now be sure he only needs to reproduce finitely many functions.
We could aswell stop doing any math, if not stop thinking.
At least stop imagining the supernatural.
The axioms that I use are just Kxy = x, and Sxyz = xz(yz).
But you allow rules of inference that permit inferences about the
enumerated array of all functions.
Brent
There is no axiom of infinity, nor even induction axiom. That belongs only to
the observers, and the proof of their existence requires only the two axiom
above, or the arithmetic one, or anything Turing equivalent. With less than
that, there is no computer, nor laptop … The universal machinery is potentially
infinite. The universal machine is finite.
Bruno
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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