🚀 go-pugleaf

zeleny@husc10.harvard.edu (Michael Zeleny) writes:
[about Penrose' argument from Godel's Theorem]

>The correct question is whether we could be working algorithmically
>without being ever being able (in principle) to attain comprehensive
>awareness of the algorithm.  For that is exactly what you would need
>in order to avoid the Lucas-Putnam-Penrose conclusion.

No, that's not correct. Even if there were some way to prove that our
reasoning was not describable by an algorithm, the conclusion that our
reasoning is more *powerful* than an algorithm simply doesn't follow.
Just because a theory is non-algorithmic doesn't mean that it is more
powerful than any algorithmic theory.

Penrose' arguments simply show that our reasoning is more powerful
than the sequence of theories T1 = PA, T2 = T1 + con(T1), T3 = T2 +
con(T2), etc. However, there are r.e. theories (for example, NF or
ZFC) that are more powerful than all of these theories.

Daryl McCullough
ORA Corp.
Ithaca, NY