On 29 May 2013, at 22:46, meekerdb wrote:
On 5/29/2013 9:52 AM, Bruno Marchal wrote:
On 29 May 2013, at 18:37, Bruno Marchal wrote:
On 29 May 2013, at 17:47, Quentin Anciaux wrote:
http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Construction_of_a_statement_about_.22provability.22
2013/5/29 John Clark <[email protected]>
On Wed, May 29, 2013 at 2:37 AM, Bruno Marchal
<[email protected]> wrote:
>> If you want to communicate why should I need to search at
all? And if even Google doesn't know what the hell Bp&p is then
it's ridiculous to expect your readers to know what you're
talking about.
> Come on, John. Search for "true opinion". Bp & p is a formula
using some notation for this,
So when I read your post and you said "Bp & p" I should have said
to myself "obviously if I Google "true opinion" it will tell me
what Bp & p means. Well, that is not obvious to me at all but it
doesn't matter because I just Googled "true opinion" and I still
can't find a damn thing about Bp & p.
Bp = I believe in p, or 'my opinion is that it is the case that
p', or, in the context of ideally correct (and simple machine):
Beweisbar('p').
p, when produced by some system, means, in all books on logic,
that p is true (from the system pov).
So Bp & p is a ay to model true opinion, in some system.
When I write I always ask myself if anybody will understand what
I say, I may not always be successful in making myself clear but
at least I try. You're not even trying.
I have explained this more than one times on this list, to
different people, because once you get it you can't forget.
You have come perhaps too much recently, but you can always ask
question. You should not focus on the formula, but on what it
represents. It is also explained in sane04, and basically, in all
my papers on this subject. Probably with different notations.
Or perhaps you just agree with what Niels Bohr said "I refuse to
speak more clearly than I think".
Bp is for "I believe p", produced by some machinery (machine,
formal system, ...).
In particular, it is an expression in some modal logic. 'Belief'
obeys usually the axioms:
1. B(p->q) -> B(p -> Bq)
2. Bp -> BBp
Bp & p means "(I believe in p) and p". P alone, in the assertative
mode of some entity means "it is the case that p". (independently
of the veracity of p).
For knowledge, we use the axiom:
3. Bp -> p
As Gödel saw in 1933, beweisbar, or provability, does not obey to
that third axiom, and so provability cannot model
knowledgeability. Indeed no consistent machine can prove B('0=1') -
> 0=1, which is equivalent with ~B('0=1'), which is self-
consistency.
I'm not sure I understand this. Are you saying we cannot take "(Bp-
>p) for all p" as an axiom because it would entail Bf ->f and then
~f->~Bf, and since ~f is true by definition it would entail that the
machine is consistent?
Yes.
More generally p -> f is equivalent with ~p, as you can verify by
doing the truth table:
p -> f
1 0 0
0 1 0
That's why Löb's theorem B(Bp -> p) -> Bp generalizes Gödel's second
incompleteness theorem: just replace p by f. B(~Bf) -> Bf, ~Bf ->
~B(~Bf), Dt -> ~BDt.
Bruno
http://iridia.ulb.ac.be/~marchal/
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