ljdickey@watmath.UUCP (10/11/83)
If one wants meaning from mathematics, one has to question the
consequences of various axioms. when something so seemingly
simple as the axiom of choice, leads to results like the
Banach-Tarski Paradax, one realizes how important it is to be
careful about which axioms one admits to their system.
I, for one, find the Axiom of Choice and its consequences
amusing, but find no value in any results that rely on them.
--
Lee Dickey, University of Waterloo. (ljdickey@watmath.UUCP)
...!allegra!watmath!ljdickey
...!ucbvax!decvax!watmath!ljdickeyljdickey@watmath.UUCP (10/12/83)
(re-submission)
If one wants meaning from mathematics, one has to question the
consequences of various axioms. when something so seemingly
simple as the axiom of choice, leads to results like the
Banach-Tarski Paradax, one realizes how important it is to be
careful about which axioms one admits to their system.
I, for one, find the Axiom of Choice and its consequences
amusing, but find no value in any results that rely on them.
--
Lee Dickey, University of Waterloo. (ljdickey@watmath.UUCP)
...!allegra!watmath!ljdickey
...!ucbvax!decvax!watmath!ljdickeyleichter@yale-com.UUCP (Jerry Leichter) (10/14/83)
Whatever your own feelings about the Axiom of Choice, most mathematicians today accept it. Banach-Tarski may SEEM paradoxical, but huge amounts of modern mathematics depend on Choice. In fact, the final result that finally got people to stop bothering, for the most part, with writing "given the axiom of choice" was the proof of its consistency: I.e. Set theory + choice (or + not choice, for that matter) is consistent iff set theory is. (I think this is a result of Paul Cohen's; it was proved back in the early '60's.) -- Jerry decvax!yale-comix!leichter leichter@yale