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.
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.
Bruno
Brent
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