On 5/18/2017 2:25 AM, Bruno Marchal wrote:
On 17 May 2017, at 20:42, Brent Meeker wrote:
On 5/17/2017 3:27 AM, Bruno Marchal wrote:
Exactly. I might try to add some possible mathematical precision,
but I need to think a bit on this. Later. Up to now, the B of Bp & p
is interpreted by its computational rendering, but "B" is really
provability, and not computation. Up to here, that absence of
distinction works well (indeed for a very deep and subtle reason
related to Mechanism), but for the precision I want to add, I will
need to make the distinction.
Are you, and others, OK with those facts:
RA cannot prove the consistency of RA. PA cannot prove the
consistency of PA, etc.
But:
PA can prove the consistency of RA.
Now the key fact which I intend to use is that RA can prove that PA
can prove the consistency of RA. In fact RA can prove also that F
can prove the consistency of PA, and of RA.
What is F?
Oops. It is ZF (Zermelo-Fraenkel Set Theory)
Despite this RA cannot be convinced that those facts prove its own
consistency (by incompleteness).
Are you going to introduce a new modal quantifier "convinced". I
already find the equivocation between B=believes, B=proves,
B=computes obfuscating.
Convinced meant "get the rational justification of". It is still "B".
Careful, I equivocate believable and provable, but not computable,
except that it happens that for sigma_1 provability is Turing
universal, and so, we can equivocate them in some context (all
computations can be shown equivalent witth proving a sigma_1
sentences). But the conceot remains intensionnally different, so this
last equivocation works only in some context and we have to be cautious.
By "believe" I made it utterly clear. M believes p means M asserts p.
Then you have to keep in mind that, in order to derive physics from
machine self-reference, we limit ourself to arithmetically correct
sound extensions of PA. In that context, Gödel's incompleteness makes
"proof" into "belief" as opposed to knowledge, because by
incompleteness the sound machine cannot prove Bp -> p.
But you've defined knowledge as true belief, not provable belief.
This means also that although( Bp & p) *is* equivalent with Bp (we
know that because we limit ourself to sound machines),
This is another confusing point. How do we know a machine is sound?
You're hypothesizing one - but there is no effective means of
recognizing one.
the machine itself cannot see the equivalence, and, indeed (Bp & p)
obeys a logic of knowledge when Bp obeys a logic of rational belief.
What does "rational" mean in that context?...provable?...or is something
like RA proving that PA can prove the consistency of RA mean that RA can
rationally believe RA is consistent...even though it can't prove it?
Real-life logic adds a non monotonic logical layer, where axioms
(beliefs) can be withdrawn, making us locally not Löbian, but that is
another story, out of the scope of mechanist theoretical physics.
So, all what I say applies to you, in the case you believe in PA
axioms, and are self-referentially correct, which you might, or not, be.
And even if I am, can I know it?
Note that the NON self-referentially correct concrete machine will
obey to the same physics
I thought you were trying to infer physics from what the machine can
prove...in sense does it "obey" physics.
Brent
than the correct one, but we can't derive physics from their
introspection, no more than we can derive history from an interview of
a guy who claim to be Napoleon.
All you need to assume is mechanism, which includes a belief in PA
axioms implicitly (if only to define what is a universal machine).
Bruno
Brent
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