Re: Tarski (complete and consistent?)

[twayburn@juno.com (Thomas L Wayburn)]

On Mon, 10 Nov 1997 09:37:08 -0500 Chuck Britton <britton@odie.ncssm.edu>
writes:
> 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 spherical 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.
>
>
> hmmm.
>
>  -.     .-.     .-.     .-.     .-.     .-.     .-.     .-.     .-.   
>  .-
>    \   /   \   /   \ N /   \ C /   \ S /   \ S /   \ M /   \   /   \  
> /
>     `-'     `-'     `-'     `-'     `-'     `-'     `-'     `-'     
> `-'
> Chuck Britton  					Education is 
> what is left when
> britton@odie.ncssm.edu  			you have forgotten 
> everything
> North Carolina School of Science & Math		you learned in 
> school.
> (919) 286-3366	x224				Albert 
> Einstein, 1936
Wayburn replies:

This reminds me of a conversation  Feynman related in *Surely You're
Joking, Mr. Feynman*.    Of course, extremely bizarre results can be
obtained if one defines the topology in an interesting way.  This oddity
is probably completely sensible in the space Tarski and Banach were
using.  We never - but never - use the actual space in which objects in
the universe are embedded; but, sometimes, we use some pretty bizarre
spaces for excellent practical purposes.  Remember, they laughed at
Lobachevsky too.  (How can a point be a line?)   But, he stands
vindicated now.    First, determine what space you are in.  (Given the
quantum weirdnesses we see AND the necessity to add dimensions to unify
the five fundamental forces (gravity, electromagnetic, weak force
(nucleus(, strong force (nucleus), and velcro) we might find ourselves
living in space a lot weirder than that.  

Actually, what even the most careful mathematicians call a proof would
not pass muster according to the precepts of Whitehead, Russell, Tarski,
and Banach.  Remember Brouwer and strict constructionism.  And you
thought you had problems!

Regards / The Dilettante

P.S.  So you couldn't help out Michigan by beating Florida State, eh?  
But, the freak accident at the end of the Nebraska - Missouri game really
hurt.  Anyway the sports writers knew which team is best.  However, we
Wolverines have had our hearts broken too many times to get our hopes too
high yet.