Re: Tarski (complete and consistent?)

[PPARKER@TWSUVM.UC.TWSU.EDU]

> Date: Mon, 10 Nov 1997 09:37:08 -0500
> From: Chuck Britton <britton@odie.ncssm.edu>
>
> twayburn@juno.com (Thomas L Wayburn) quotes:
> > If the axiom system of a deductive theory is complete, and if any
> > sentence which can be formulated but not proved within that theory is
> > added to the system, then the axiom system extended in this manner is no
> > longer consistent. - Tarski, Alfred, *Introduction to Logic ...*, Oxford,
> > New York (1994), p. 133.
>
>
> And anyone who is 'into' axiomatic mathematics needs to read up on the
> Banach-Tarksi results that clearly show that our current understanding of
> differential calculus leads to the result that a shperical set of points
> with size equal to the sun can be dissected into a finite number of subsets
> that can be reassembled into a sphere the size of a pea (or any other size)
> with no points left out, overlapping etc.

Banach-Tarski is purely topological.  In fact, it *cannot* be done
differentiably, and that's one of the major reasons we prefer smooth things
to those that are merely continuous. (Another fine example is the existence
of space-filling curves; they can't be smooth either.)

---------------------------------------------
Phil Parker        pparker@twsuvm.uc.twsu.edu
Random quote for this second:
  But in modern war...you will die like a dog for no good
  reason.---Ernest Hemingway