Russell Standish wrote:
>Interesting, although I suspect the interpretation of "the ability to
>do something completely stupid" is more like asserting the truth of
>an unprovable statement than asserting the truth of a false
>statement. In modal logic, this would be (x & -x ) n'est-ce pas?
I would say it is asserting the possibility or consistency (instead of
the truth) of some false but irrefutable statement: like <>f, which
belongs to G* minus G, which means <>f is indeed a solution of
x & -x, that is G* prove <>f & -<>f. Note <>t is also a solution
of x & -x
Of course the statement f is false on the sound machines by
definition! But <>f is true. The correct machine "suffers" from the
consistency of inconsistency.
>Note an automaton cannot assert the truth of anything not provable
>from its axioms...
An "automaton" cannot even assert the truth of its provable assertions
in general. Except the one which has been proved. (Lob), so, ok.
But an automaton can *anticipate* or bet on the possibility
(the consistency) of both what he/she/it can prove (G) and even some
truth he/she/it *cannot* prove (which truth can works as fruitful
questioning or betting or hoping or fearing ...) (G*).
Or even quantum probabilising on. (Z1*).