On 06 Jul 2017, at 00:08, John Clark wrote:
On Wed, Jul 5, 2017 at 11:51 AM, Bruno Marchal <[email protected]>
wrote:
>> the study of the largest prime number seems rather silly
to me.
> It is just a logical point that you can't derive this from
Robinson Arithmetic theory,
That doesn't bode well for the future prospects of the Robinson
Computer Hardware Corporation now does it.
? If RA did not exist, or did not make sense, computer would not
exist, neither in arithmetic, nor in any reality.
> a bit like the Euclid postulate on parallel has been shown
non derivable from the other geometric axiom.
But neither Euclid's Fifth Postulate nor its absences
leads to a world where logical contradictions can exist, but a
finite number that is the largest prime does.
We cannot define "finite" in first order logic, and RA + "there is a
largest prime" is a consistent theory. The situation is exactly like
what happens with the Euclid's Fifth Postulate. That is the point. RA,
despite being already Turing universal, and able to do what any Turing
machine can do, is consistent with the existence of a biggest natural
number.
>> A Turing machine can do Peano Arithmetic too.
Nothing new in that.
> Of course not,
Of course not??
A Turing machine can emulate PA, like it can emulate Einstein
(assuming mechanism). But there are no reason a Turing machine would
believe in what PA believes, or in what Einstein believes. Now, the
word "do" is rather loose, and I took it that a Turing machine
believes in PA axioms, which is not generally the case.
> I said that I wrote different program for a Löbian machine,
and if you read Smullyan's book, you will find others, and they are
all emulable by a Turing machine
If it can emulate it then whatever a "Löbian machine" is
and whatever it can do a Turing machine can do it too,
including Peano Arithmetic. As I said, nothing new.
Yes, but that is trivial. A Turing machine can also emuate "PA +
inconsistent(PA)" (which is a consistent theory). Nothing new, but you
were confusing "doing" and "beliving", or "computing" and "proving".
> you seem to ignore the difference between a theory, or a set
of "believed" propositions/sentences. A (universal) Turing
machine can compute (everything computable), but cannot prove
anything.
As far back as 1956 a computer had proved 38 theorems in
Whitehead and Russell's Principia Mathematica, and some of the
proofs were judged by most to be more elegant than the proofs
Russell and Whitehead found. And the old machine used to do
it was equivalent to a Turing Machine, as are modern computers.
>> Turing did far more than define what his machine could
do, he explained exactly how to construct one in great detail, and
that's why other people gave it the name "Turing Machine".
> Yes, he is the discoverer of the (mathematical) notion of
computer, and he was interested in building one.
It's true Turing built actually electronic devices but what I
meant was in in his original article about a long paper tape and
marking pen and a eraser etc he described exactly how to build one;
he didn't expect anyone would actually build a practical machine
that way but he did prove it was physically possible to do so. Where
is your equivalent for a Löbian machine?
There are many in the literature, in biology (human brains), and it is
easy to implement them. Boyer and More made nice machine/programs
explicitly proving their own löbianity (in the present sense of my
previous post).
Anyway, if we are machine, we have to extract physics from Löbianity,
and for this all you need is a pen and paper.
Martin Löb knew exactly what is a Löbian machine, although he would
have called it a "sufficiently rich theory"
I know what departments in my big box hardware store to go to to
buy light detectors, paper tape, erasers, marking pens, and gears
and pulleys to build my Turing Machine; but what department sells
"sufficiently rich theory" and how heavy is it, will I need help
carrying it to my car?
> Like I said, PA cannot prove its own consistency, but PA can prove
that ZF can prove PA's consistency,
That doesn't solve the problem,
Which problem? Nobody doubt that RA is consistent, even Nelson. Nobody
doubt that PA is consistent (except Nelson).
it just kicks the problem down the road.
The problem of consistency is solved, in the negative, by Gödel. We
cannot do that. We can still be pretty sure that some theories are
consistent, but it makes no sense to try to prove it, except in
metamathematics, to illustrate some principle.
Is Zermelo–Fraenkel consistent? ZF can't say, and even if it
is there are true statements that is can't prove.
That was my point. PA can prove that ZF proves PA's consistency, but
that is not a proof of consistency from PA's point of view.
Bruno
John K Clark
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