[FOM] Finitist concept of consistency?

[Stephen Pollard]

I'm interested to see how FOM'ers respond to an old question about  
Hilbert's program. This could be very helpful to a couple dozen  
undergraduates who will be tackling the question in a few weeks.

Suppose I have a mechanical procedure for enumerating discourses I  
call "S-proofs." Suppose I manage to show that, say, proof #17 is not  
a proof of absurdity, but I do this in an odd way. I don't actually  
look at proof #17. I reason simply from the fact that #17 is an S- 
proof. I recognize that this reasoning can be mechanically adapted to  
show that proofs #289, #64, and some other proofs I choose at random  
are not proofs of absurdity. This adaptation of the reasoning does  
not rely on any special features of the chosen proofs. All that  
matters is my recognition that they are S-proofs. I have now learned  
something important about S-proofs. I don't want to forget my  
insight, so I write down two reminders.

1. Proof #n is not a proof of absurdity.
2. The formal theory S is proof-theoretically consistent.

Suppose I have done all this without any quantification over an  
infinite domain (either in my assertions or in the content of my  
beliefs). Suppose, too, that there is no upper bound on the length of  
S-proofs. Now here's the question. Have I recognized that the formal  
theory S is consistent? If I have not represented to myself the  
proposition that S is consistent, then I have not recognized that S  
is consistent. Can I represent this proposition to myself without  
adverting to all S-proofs, where the "all" ranges over infinitely  
many things?

As I understand it, Oskar Becker gave the answer "No" back in the  
1920's. In his 1981 paper on finitism, Prof. Tait argues that nothing  
in principle prevents finitists from verifying quantifier-free  
formulas that non-finitists would interpret as universal claims about  
the natural numbers. This should help us answer the question of what  
finitists are in a position to assert when they enunciate such a  
formula. I am not confident, however, that I understand what that  
answer would be under Prof. Tait's analysis.


Stephen Pollard
Professor of Philosophy
Division of Social Science
Truman State University
spollard at truman.edu