FOM: Standards of mathematical rigour and logical consequence

Charles Silver csilver at
Mon Nov 2 15:37:08 EST 1998

C. Silver wrote:
> >         I find your finitistic viewpoint very interesting, but there are a
> > couple of things I don't understand....

Vladimir Sazonov answered:
> Before a set theory and then model theory and Goedel 
> completeness theorem arose, say, in the time of Euclid there 
> were no *set-theoretically defined* logical consequence 
> relation. There were essentially syntactical (in a broad sense 
> of this word) rules of "correct" reasoning.  Newcomers learned 
> these rules by training, i.e. as given and, of course by some 
> appeal to geometrical and other intuition.  These rules arose 
> (due to also some peoples, professional mathematicians) 
> according to and SIMULTANEOUSLY WITH creating this intuition.  
> Each newcomer simply repeat in a shorter way this creation 
> process with the help of a teacher. But he, of course, more 
> learn, sometimes even grind than create himself. 

	Thank you for your long post.  I think I now understand your
viewpoint better.

> Is FOL 
> "really" complete? (Cf. also my paper in LNCS 118 (1981)).  May 
> be we "really" should have a kind of incompleteness of FOL?  Or 
> should/can we just consistently(?) *postulate* completeness 
> which also seems plausible and is very desirable? 

	I would like to understand in what sense FOL is not "really"
complete.  Could you please furnish the title and page numbers of your
article?  Thank you.

Charlie Silver

More information about the FOM mailing list