Stop trolling.
Bruno
On 06 Jul 2017, at 19:54, John Clark wrote:
On Thu, Jul 6, 2017 at 11:10 AM, Bruno Marchal <marc...@ulb.ac.be>
wrote:
>> 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,
Then first order logic is not of much use in the physical
world, except perhaps as a toy logicians can use as training
wheels until they get good enough to tackle real problems.
> The situation is exactly like what happens with the Euclid's
Fifth Postulate.
No it is not exactly like that. Euclid's Fifth
Postulate (and the continuum hypothesis too) is either
true or it's not, and neither possibility leads to a logical
contradiction. A largest prime number also either exists or it does
not, but if it does exist then logical contradictions do too.
> A Turing machine can emulate PA, like it can emulate Einstein
Yes, and thats only one reason why it's such a powerful machine.
> But there are no reason a Turing machine would believe in
what PA believes, or in what Einstein believes.
Well, did Einstein believe in anything or did he just write
symbols on paper that got published as journal articles? If you
think he did believe it I'd like to know how you determined that,
and then I want to know what exactly Einstein's brain had that the
Turing Machine (the one that was doing such a good job emulating
Einstein) lacked.
> 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.
Did Giuseppe Peano believe in the Peano Postulates, or did
he just write symbols on paper that got published as journal articles?
>>> 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.
TRIVIAL?! I couldn't fail to disagree with you less.
> A Turing machine can also emuate "PA +
inconsistent(PA)" (which is a consistent theory).
The more things a Turing Machine can emulate the more powerful it
is.
> Nothing new, but you were confusing "doing" and "beliving",
or "computing" and "proving".
And you're confusing truth and proof.
>> 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 a Turing machine can emulate a human brain, and you say
that's a "Löbian machine", if so then in 1935 Turing
showed exactly how to build a "Löbian machine",
whatever the hell that's supposed to be.
> PA can prove that ZF proves PA's consistency, but that is not
a proof of consistency from PA's point of view.
Point of view? You may or may not be able to prove what category
something belongs in but all statements are either true or false or
gibberish.
John K Clark
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