FOM: Standards of mathematical rigour and logical consequence
csilver at sophia.smith.edu
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
> 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.
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