[FOM] Infinite proofs
praatika at mappi.helsinki.fi
Wed Aug 22 08:17:44 EDT 2007
Alex Simpson <Alex.Simpson at ed.ac.uk>:
> Presumably what you mean is that proofs are well-founded
> trees, so every branch is finite. Nonetheless, such proofs still
> have, in general, an infinite "height" given by a countable ordinal
Surely! That is why I think they are, in an important sense, infinitary.
> Since you do not seem happy with well-founded proofs, perhaps
> you are looking for systems involving non-well-founded proofs.
Actually, all this has nothing to do with me liking or disliking
such "proofs". I only tried to reply Ron Rood's request about "infinitely
long proofs" and first just wanted to point out Hazen's admirably
accessible survey. 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 proofs might be like...
It is interesting if there are also such non-well-founded "proofs" in the
logical literature. I am grateful for all the references.
It should be only remembered that all such infinitary "proofs", whether
well-founded or not, are EXTREMELY different from what is usually meant by
a "proof", i.e., where one can always effectively check whether an alleged
proof really counts as a proof or not.
Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy
Department of Philosophy
University of Helsinki
E-mail: panu.raatikainen at helsinki.fi
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