On 29 Nov 2017, at 23:10, Brent Meeker wrote:
On 11/29/2017 6:36 AM, Bruno Marchal wrote:
On 29 Nov 2017, at 02:19, Brent Meeker wrote:
On 11/28/2017 7:06 AM, Bruno Marchal wrote:
On 27 Nov 2017, at 23:02, Brent Meeker wrote:
On 11/27/2017 10:19 AM, Bruno Marchal wrote:
The formal modal formula is B(Bp -> p) -> Bp.
It looks also like wishful thinking. If you succeed in
convincing a Löbian entity (whose beliefs are close for the
Löb rule, or having Löb's theorem for its bewesibar predicate)
that if she ever believes that some medication will work, then
it work, then she will believe the medication works!
But that's equivocating on "B". In the formula it means
beweisbar=prove not "believes". I think that is obfuscation.
Before Gödel, everyone thought that B was an operator for a
knowledge predicate. But after Gödel (1931), we know that B
verifies all the axiom of knowledge minus the key one (Bp -> p),
making it into a (rational) notion of belief. I could have
defined by axiomatically belief by:
The subject believes x + 0 = x, the subject believes x + s(y) =
s(x + y), etc.
But the etc. involves infinitely many beliefs...which I don't have.
No, you need only the finite number of beliefs:
I think the belief that there is a total successor function s() is
already and infinite set of beliefs.
You mean that you belief in 0, in s(0), s(s(0)), ... me too. It is a
potential infinite, and it is nameable through the code of s, which is
provably total by PA.
We need such belief to even believe in notion like circle. But it is
not part of the ontology.
RA cannot prove that, but it is still true (= satisfied by the
structure (N, 0,, +, *).
I think you belief in Schroedinger equation, which involves bigger
infinities. RA (the ontology) has no axiom of the infinite, and when
you talk about the infinity of numbers, or of beliefs, you are at the
meta-level, which in this case will be played by PA, and some
extensions (like ZF). RA can proves their existence, and we will study
the personal diaries of PA and ZF, when getting new axioms, as long as
they keep arithmetical soundness.
PA can prove assertion which are equivalent to "infinity statement"
without involving infinity, like PA can prove that for each natural
number, there is a bigger one. PA can prove, in that sense, the
infinity of prime numbers, like Euclid.
bruno
Brent
0 ≠ s(x)
s(x) = s(y) -> x = y
x = 0 v Ey(x = s(y))
x+0 = x
x+s(y) = s(x+y)
x*0=0
x*s(y)=(x*y)+x
Together with the belief that if ever you believe A, and A -> B,
you will believe B.
Such beliefs can be distributed in a neural net on with any finite
representation. Here we make things simple.
If this applies to you, as I am sure it does, the results will
apply to you or any of your recursive computational continuations.
This would be ridiculous if that was used to model human
psychology,
Then why do continually use the word "belief" which does refer to
human psychology? I think you are obfuscating the assumption that
your "ideal entities" "believe" everything provable from whatever
set of axioms characterize them.
Yes. That is why sometimes I use "rational belief". In fact, I
could define belief by assertion. I could say that a machine
believes A if she asserts A, and then explain that I limit myself
to machine which never believes an arithmetical falsity. So the
machine could have any number of non-arithmetical axioms, with the
condition that this makes not them inconsistent in arithmetic. If
you believe in the axiom above, and in the closure of the axioms
for the modus ponens rule (that means that the modus ponens
preserves the truth of the axioms) then, if you would say yes to a
doctor and survive, the consequence of Löbianity apply to you, and
explain why the observable obeys a quantum logic, with apparent
interfering multi-histories.
Bruno
Brent
but it is not a problem for the derivation of the physical laws
(unless you believes that the universe depends conceptually of
human psychology, but that would be a rather strong coming back
to the kind of anthropomorphism we usually avoid here.
Bruno
Brent
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