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
> consciousness.


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.

Bruno

http://iridia.ulb.ac.be/~marchal/




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