On 6/4/2020 3:39 PM, Lawrence Crowell wrote:
On Thursday, June 4, 2020 at 6:07:45 AM UTC-5, Bruno Marchal wrote:
> On 2 Jun 2020, at 19:34, 'Brent Meeker' via Everything List
<[email protected] <javascript:>> 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] <javascript:>> 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, 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
Of course any computation is going to be finite or involve a finite
number of bits. This happens as well with quantum computers, but there
is one difference. Two states can be prepared and entangled so they
have a continuum of probabilities depending upon measurement angle.
This is what separates QM from classical mechanics.
I've wondered about this. Of course a lot variables in the theory are
continua; not just angle but also position. Yet none of those can be
measured to arbitrary precision. And the more precisely one is, the
less precisely it's conjugate can be...which is what separate QM from
classical mechanics. Holevo's theorem limits what we can know about a
state.
Brent
This separates entanglement of spins from the Bergman's socks, where
knowing the left sock is in one box the right must be in the other. So
while there is a finitude to the entanglement entropy or the quantity
of quantum information, the possible ways an entanglement can register
outcomes is infinite. This is what gives a violation of Bell's theorem
in QM. With the measurement of a quantum system the pair of a state
and measurement forms a type of Godel numbering. This connects QM
foundations with the phi_u(x, y) = phi_x(y),you state above.
A classical computer will always be finite, and you can't have an
infinite Cantor diagonalization. The computers that are manufactured
are done so to solve certain problems, RSA encyrption, user interfaces
for service personnel from travel agents to sales, word processors,
games, cell phone signal shifters, data processors of medical
measurements and on it goes. Even with quantum computers this will
take off, and in fact I have thought quantum computing would be a way
of managing a dynamics network defined by millions of drones over a
city. Even if as I think the Godel-Turing result underlies
obstructions between entanglement types quantum computers will in time
become the province of engineering and business applications.
LC
>
> 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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