FOM: Standards of mathematical rigour and logical consequence

[Charles Silver]

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.

V.S: 
> 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