Bruno Marchal wrote:
> On 23 May 2009, at 09:08, Brent Meeker wrote:
>> But why? Why not RA without induction? Is it necessary that there be
>> infinite schema? Since you phrase your answer as "I am willing..." is
>> it a matter of your intuition or is it a matter of "degree" of
> OK. I could have taken RA. But without the induction axioms, RA is
> very poor in provability abilities, it has the consciousness of a low
> animals, if you want. Its provability logic is very weak with respect
> to self-reference. It cannot prove the arithmetical formula Bp -> BBp
> for any arithmetical p. So it is not even a type 4 reasoner (cf
> Smullyan's Forever Undecided, see my posts on FU), and it cannot know
> its own incompleteness. But it can be considered as conscious. It is
> not self-conscious, like the Lobian machine.
> Note that Bp -> BBp is true *for* RA, but it is not provable *by* RA.
> Bp -> BBp is true for and provable by PA. Smullyan says that PA, or
> any G reasoner, is self-aware.
> Of course, consciousness (modeled by consistency) is true for PA and
> RA, and not provable neither by RA nor PA (incompleteness).
> But all this is not related to the problem you were talking about,
> which I still don't understand.
I think it is related. I'm just trying to figure out the implications
of your theory for the problem of creating artificial, conscious
intelligences. What I gather from the above is that you think there are
degrees of consciousness marked by the ability to prove things. To
consider another view, for example, John McCarthy thinks there are
degrees of consciousness marked by having narratives created and
remembered and meta-narratives. Either of these ideas is definite
enough that they could actually be implemented (in contrast to many
philosophical ideas about consciousness). I have some reservation
about your idea because I know many people that I think are conscious
but who couldn't prove even the simplest theorem in PA. Are we to
suppose they just have a qualitatively different kind of consciousness?
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