On 19 Dec 2012, at 20:18, Stephen P. King wrote:
On 12/19/2012 2:14 PM, Bruno Marchal wrote:
I am trying to see if we can use the way that towers of theories
are allowed by the incompleteness theorems...
This is studied in recursion theory. Turing shows that
incompleteness continue to all effective transfinite tower, on the
constructive ordinals.
Dear Bruno,
Yes, but we can see relative completeness between neighboring
levels of the tower, no? Statement in Theory A that cannot be proven
in A can be proven to be true in theory B that includes and extends
beyond theory A, no?
That is why they are towers, yes. But even the union of tower remains
incomplete, unless the extension are done in an effective way, in
which case we build more "models" than "theories" in the usual sense.
Bruno
--
Onward!
Stephen
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