[FOM] Infinite proofs

[Ron Rood]

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