On 26 Sep 2015, at 23:36, John Clark wrote:
On Sat, Sep 26, 2015 Bruno Marchal <marc...@ulb.ac.be> wrote:
>>Paul Cohen not Godel proved that arithmetical reality is
independent of the Axiom of Choice
> I don't think so. The independence of arithmetic from AC in
ZF follows from Gödel's proof that V=L -> AC. A model of ZF where
all sets are "constructible" (V = L) verifies the choice axiom
If a axiom has been verified, that is to say if it can be derived
from other axioms,
I meant verified in a model, not prove in a theory. If a proposition
is verified in a model, its negation can still be verified in another
model. That proves only the consistency of the proposition.
then it no longer needs to be a axiom and is just the result of more
fundamental axioms. Paul Cohen proved that AC can not be
derived from ZF.
>>Godel proved that if you assume that AC is true ZF will
produce no contradictions, 25 years later Paul Cohen proved that
if you assume AC is false ZF will STILL not produce any
contradictions, and so AC must be independent of ZF and can not be
derived from ZF.
> Yes, but this has nothing to do with what I am saying. The
fact that the arithmetical truth is independent of the choice axiom
Then a lot of stuff that mathematicians think is true is not true,
or at least can't be proven to be part of "arithmetical truth
" because the Axiom of Choice is needed to prove them.
The constructible set of Gödel can be use to show that ZF and ZFC
proves the same arithmetical theorems. But of course richer theory can
prove more theorem. ZF proves much more than PA, and ZF+kappa proves
much more (purely arithmetical) proposition than ZF. It is
inexhaustible. No axiomatisable theory at all proves all arithmetical
propositions. Arithmetical truth is just not axiomatizable. That
follows from Gödel's incompleteness or from theorem by Skolem, etc.
> can be seen as a corollary of Gödel's proof that AC is
consistent with ZF,
Godel proved in 1938 that AC was consistent with ZF but for all
Godel knew The Axiom of Choice could be derived from Zermelo-
Fraenkel; and that is in fact what Godel believed at the time and
what most mathematicians thought, even Paul Cohen thought so
and was as surprised as anyone when he found in 1963 that the
negation of AC was consistent with ZF too and thus independent of ZF.
>>> Physics is a theory about a possible physical reality
>> I know. So if physical reality is ZFC ( a big "if"
I admit but it could be) then physical reality has something that
arithmetic derived from just ZF does not have.
>> "physical reality is ZFC" means nothing to me.
Physical reality is Zermelo-Fraenkel plus the Axiom of
Choice, "arithmetical truth" is just Zermelo-Fraenkel.
This is a bit of non-sense. ZF see only a fragment of the arithmetical
truth, and "physical reality" is a god in which I tend to be rather
skeptical about.
I can explain why it makes no sense once we postulate
computationalisme, but you need to grasp the UD argument step 3 to get
this.
I'm not saying it's true, I'm just saying that's what it means; it
might be wrong but it's not gibberish.
> Anyway, I do not assume that there is a physical reality.
Hmm. Margaret Fuller once said "I accept the universe" to which
Thomas Carlyle replied "Gad, you'd better". Unlike you at
leas Fuller accepted the universe, I wonder what Carlyle would say
to you.
I believe in a physical reality, but not in one that we have to
assume. I can prove that physicalism is meaningless once we bet the
physical brain can be emulated by a physical Turing machine.
>> And yet despite repeated requests you are unable or
unwilling to explain why you can't start the Tiny Sigma_1
Computer Hardware Corporation and become the richest man on the
planet.
> Not at all. I think you don't read the answer. The
answer, I repeat again, is that to build an hardware corporation I
need hardware
Yes perfectly true, you need physical hardware. But my question is
WHY? The only answer can be that physical hardware has something
that "arithmetical truth" does not.
Then you artificial brain is not Turing emulable, and computationalism
is false.
We may not be certain what that something is but the fact that
computer hardware companies have non zero manufacturing costs is
proof that one has something the other does not.
No, because that relative cost exost also in arithmetic reelatively to
the people emulated in arithmetic. There too some John Clark pretends
there is a physical universe, and we know he is wrong, even when it
hits on the table and say "looks that is hardware".
> and I need to implement the universal machine in that
hardware.
Yes exactly you need to implement it, but to implement it
mathematics needs help, it needs physics!
We agree on this, but that is not a proof that hardware exist, nor
that we have top assume it to explain it.
> to have a computation running, I need only arithmetic.
But it is a fact that to have a successful company that provides
answers to arithmetical problems arithmetic is NOT all you need.
> Numbers ==> computations ==> dreams ===> physical reality
===> physical computation ===> hardware company
OK, but the hardware company certainly has access to numbers so
why doesn't INTEL just make calculations directly and forget about
all that unnecessary and expensive messing around with silicon?
Because if we want to share computations, we need to implement them in
the first person plural reality that we share to begin with. but that
reality is itself emerging from infinitely many computations in
arithmetic (if comp is true, and my derivation is correct).
>> You can't explain it but I can, you can't do it because a
physical silicone microprocessor chip has something that Robinson
arithmetic and "the tiny sigma_1 part of the arithmetical
truth" lacks.
> Yes, that is right. But that things which is lacking is an
illusion,
An Illusion is a perfectly respectable subjective phenomenon, and
so is consciousness; so you're saying that subjective
phenomenon is the thing that that matter that obeys the laws of
physics can create that arithmetical truth can not create.
I cannot parse this sentence.
Well maybe, arithmetical truth is certainly lacking something that
physics has.
That is a theorem in machine's theology.
You need to understand that incompleteness makes the following
modalities (poits of view) being extensionnaly equivalent (they all
see the same (recursively enumarable) part of the arithmetical truth
"at each instant"), yet they obey quite different logics, yet all
emulable by the logic G, at the propositional level).
p (arithmetical truth (at first, later replaced by the sigma_1
arithmetical truth). The simple neoplatonist God, or One, easily
associable when we assume computationalism, the believe that a right
computations can makes it possible for a first person to manifest
itself in the usual relative way.
[]p (provability by the Löbian machine, that is mainly the machine
which believes that if a natural number exists that verifies some
verifiable property, they can find it (we are allowing some time/
number of computation steps, the machine does not need to be aware of
that).
[]p & p (by the second incompleteness theorem, the machine cannot
prove ~[]f, that is she can not prove []f -> f, so she can't prove in
general
[]p -> p, and this makes the logic of []p & p differing from the logic
of []p. ([]p & p) -> p trivially). What the machine proves is only its
opinion, but []p & p, gives a logic of what the machine proves
attached by definition to truth. It is basically Theaetetus'
definition of knowledge, and here it gives indeed a modal logic of
knowledge, enriched by a special new axiom (Grzegorczyk []([] (p ->
[]p) -> p) -> p). It is the mathematical definition of the first
person associated to the machine. The theory is well known, it is
S4Grz. With comp, we will need to still slightly extend it to S4Grz1
(adding
p-> []p as axiom for the atomic formula p)
[]p & <>t, that is the logic of the one who say yes to the doctor,
without asking for a proof, as he knows that if he survives, he will
still unable to prove the fact.
As the machine used talk first order language, <>t is equivalent with
the existence of a model. With the first person modality above, we
attach the believer/prover to the truth. With the "& <>t" we attach
the believer with a possible model/reality/universe/god/whatever-
realizing-me. That is a consequence of Gödel's completeness theorem.
It is an implicit religious believer who does not believe in cul-de-
sac worlds. It is thus also the logic of the measure one: []p = p is
true in all model, and "& <>t " and this means something as there is
at least one world accessible (I am not in a cul-de-sac world).
That is the first person plural observable, at least its modal logic.
It inherits the splitting of G to G*.
Then you can apply the Theaetetus again, and incompleteness keeps
making the modal logics consistent and different: []p & <>t & p.
When you extend the arithmetical interpretation from propositional
modal to quantified predicate modal logic, you get theories which are
highly undecidable, where even the whole set of arithmetical truth,
used as an oracle can answer question without doing infinities of steps.
Note that G and G* applies also at the analytical level, or for set
theories. It is very general, and it applies provably to a notion of
ideally self-referentialy correct "platonist" machine (here
"platonist" means they believe in Aristotle idea that a reasonable
proposition on a number is true or false)).
The phsyical is defined by what the universal machine (defined in
arithmetic) can observe, and observation is defined by "measure one on
the set of sigma_1 true sentence". So we derive physics by extracting
the logic of the observable, and if we get a quantum modal logic close
enough to the logic B, we can reverse a transformation due to
Goldblatt to get an arithmetical quantum logic, at the place we were
asking the measure one.
> like someone can emulate Einstein's brain
Then that emulation is Einstein.
Better: that emulation makes it possible for Einstein to manifest
itself.
> making a course in GR without any understanding of GR.
Then Einstein didn't "understand" GR
No, you confuse the level. The guy who manipulate the pages of the
book can talk with Einstein, but it does not become Einstein by
emulating it!
RA can emulate PA and ZF, but RA knows about noting compare to PA and
ZF. PA can prove that RA is consistent. Like ZF can prove that PA is
consistent.
And RA can emulate PA doing a proof of the consistency of RA, but RA
will not have any reason to trust PA about that, and that emulation is
not a proof of the consistency of RA by RA (which would made RA
inconsistent by the second theorem.
If you emulate Einstein, you talk with Einstein, you don't become
Einstein.
Caution; grave confusion of level.
and like "God" and more recently "theology" the word has lost all
meaning. This destruction of words you're engages in is getting
scary, pretty soon we'll just have grunts.
I use the word in the sense of the dictionnary which are aware that
there are many different religions, comparative theologies.
I use "theology" in the sense of Proclus, I mean in the sense of the
domain of free inquiry it was before politics get mixed and steal the
idea with the help of the charlatan and the credulity of the people,
exploiting fears, etc. Theology is a science. Math and physics are
born from it, but unless you justify it from some theological
hypothesis, you can't confuse theology with any science in different
domain.
>> I'm not sure exactly what it's lacking, maybe it's the Axiom
of Choice and maybe it's something else,
> It is the primitive matter which is lacking.
Then primitive matter is more fundamental than arithmetic.
QED.
OK, but then we are not Turing emulable, and you need to explain me
what magical thing, or actual infinite, you are using for that
primitive matter to select the computations, or just abandon comp, and
revised the contract asking for them to keep intact the actual
infinities in the primitive matter of your brain (good luck explaining
them what you mean).
>> but it sure as hell is lacking SOMETHING because nobody has
been able to start a computer hardware company with zero
manufacturing costs.
> But the hardware and the primitive matter are explained in RA,
Can RA also give an answer that INTEL stockholders would accept to
explain why shutting down all their silicon chip fabrication plants
and just ordering their employees to meditate about numbers didn't
turn out to be a wise business move?
RA cannot do that anymore than you can make a pizza by solving
Everett Dewitt Wheeler Universal Wave Equation.
Not sure you are trying to be serious here, or you keep confusing many
levels.
Bruno
John K Clark
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