[FOM] Infinite proofs
Ron Rood
ron.rood at planet.nl
Thu Aug 23 15:22:20 EDT 2007
praatika at mappi.helsinki.fi wrote on 22 aug 2007 at 14:17
(Europe/Amsterdam):
> But I then started to wonder whether omega-rule etc.
> really are exactly what Rood was asking for. And I was wondering a bit
> about what such [infinite] proofs might be like...
First of all thanks to all who have responded to my question.
What motivated my original request are certain constructive proofs. I
mean, for example, certain proofs for the existence of space filling
curves as provided by Hilbert, Sierpinski, among others.
These proofs proceed by the construction of a uniformly convergent
series of curves in the unit square; the limit of such a series is a
curve meeting every point in the unit square, i.e., a "space filling
curve."
Proofs like these clearly have an infinitistic flavor in that they
proceed by the construction of a (countably) infinite series of
objects. The limit of the series is the desired object.
I was wondering whether there is an analagon of such infinary proofs in
terms of logical derivations.
So, in this respect, the omega-rule does not seem to be exactly what I
was looking for since the omega-rule, so to speak, "grabs" an infinite
lot of sentences together and accordingly yields another sentence in
one single step. In view of the above, non well-founded derivation
trees are I believe neither like the examples I originally had in mind.
Any further thoughts?
Ron
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