[FOM] Infinite proofs

praatika@mappi.helsinki.fi 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.

Best,  Panu   

Panu Raatikainen

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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