On 23 Oct 2013, at 23:42, meekerdb wrote:

On 10/23/2013 5:53 AM, Bruno Marchal wrote:

On 22 Oct 2013, at 19:01, meekerdb wrote:

On 10/22/2013 5:48 AM, Bruno Marchal wrote:

On 21 Oct 2013, at 20:07, meekerdb wrote:

On 10/20/2013 11:12 PM, Russell Standish wrote:
On Sun, Oct 20, 2013 at 08:09:59PM -0700, meekerdb wrote:
Consistency is []p & ~[]~p. I was saying []p & ~p, ie mistaken belief.
ISTM that Bruno equivocates and [] sometimes means "believes" and sometimes "provable".

And I'm doing the same. It's not such an issue - a mathematician will
only believe something if e can prove it.

But provable(p)==>p and believes(p)=/=>p, so why equivocate on them?


Both provable('p') -> p, and believe('p') -> p, when we limit ourself to correct machine.


(And incidentally mathematicians believe stuff they can't prove all the time - that's how they choose things to try to prove).

Then it is a conjecture. They don't believe rationally in conjecture, when they are correct.

They believe it in the real operational sense of "believe", they will bet their whole professional lives on it. How long did it take Andrew Wiles to prove Fermat's last theorem? Since one can never know that one is a "correct machine" it seems to me a muddling of things to equivocate on "provable" and "believes".

On the contrary. It provides a recursive definition of the beliefs, by a rational agent accepting enough truth to understand how a computer work.

We can define the beliefs by presenting PA axioms in that way

Classical logic is believed,
'0 ≠ s(x)'  is believed,
's(x) = s(y) -> x = y'  is believed,
'x+0 = x'  is believed,
'x+s(y) = s(x+y)' is believed,
'x*0=0' is believed,
'x*s(y)=(x*y)+x' is believed,

and the rule: if "A -> B" is believed and A is believed, then (soon or later) B is believed.

But the point of Seth Lloyd's paper was that later can effectively be never and since given any time horizon, t, almost all B will not be believed earlier than t.

But Seth Loyd assumes some physical universe. In the arithmetic context from which we start (and have to start by UDA, at some recursive equivalence) soon or later means "once". It never means "never".


So really you call it "believe", but you use it as "provable".

You miss the point. The incompleteness forces the provability logic to behave like a believability logic. After Gödel, provable (which was for many the best case of knowledge) becomes "only" 'believable'.

That's why I agree with Popper, that science = only belief, because the big difference between a belief and a knowledge, is that the first is corrigible and the second is incorrigible.

(Popper and Deutsch uses non-standard vocabulary here, but I agree with them).






Then the theory will apply to any recursively enumerable extensions of those beliefs, provided they don't get arithmetically unsound.

The believer can be shown to be Löbian once he has also the beliefs in the induction axioms.

Not really. You have add another axiom that the believer is correct.

Why would I need to do that? It is not a new axiom, it is that I limit the interview to correct machines. (Everett does the same, it is natural. If you predict that a comet will appear in the sky, you will not be refuted by a paper explaining that when astronomers are sufficiently drunk, they miss to see it. You don't have to assume that the observer is not drunk, sane of mind, etc. (At the level of the scientific paper, in real life you know that a talk after dinner, at some conference, will count for nothing, as people are full, and sleepy!).




He doesn't believe any false propositions - which means it is an idealization that doesn't apply to anyone.

To derive physics, that is enough. Theoretical approach starts from the simpler assumptions, and change them, or improves them only if needed. If not, you would have rejected Newton's at once, as he consider the sun being a point, when recovering Keepler laws from his gravitation theory.

The interesting happening, I think, is that by G* proving <>[]f, somehow, the laws of physics and the whole machine's theology have to take into account the consistency of incorrectness, at some basic fundamental level. The idealization makes justice itself of your remark, somehow.

Bruno




Brent


Bruno


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