FOM: My BT post again
David Ross
ross at math.hawaii.edu
Fri Dec 12 10:40:06 EST 1997
OK, let me try again. The Hausdorff paradox is a paradoxical
decomposition of the sphere (as opposed to the ball) by means of
rotations; passage from the Hausdorff paradox to BT is a trivial
projection argument. The Hausdorff paradox, in turn, is proved in
basically two steps, call them A and B. The 'meat' of step A is showing
that F2 lives in the rotation group, and requires no choice. Step B is
almost identical to the usual equivalence class proof for a
nonmeasurable subset of [0,1], and of course uses AC. Since F2 is not
abhorrent, we wouldn't use step A to argue for restricting to Borel
sets. Therefore, if BT is an argument for adopting this mathematical
ontology, then the argument must already exist in step B, and therefore
in the nonmeasurable set example. *This* was my indictment of the use
of BT as a motivating example - it is overkill. (Surely a
reverse-mathematics person should be sympathetic to this kind of
objection!) If I shoot someone, then hang a really impressive "He's
Dead!" sign on the corpse, this isn't be a better argument for gun
control than the corpse alone (though of course it makes for better
television).
As for references, I think the nicest exposition of BT and related
paradoxes is still Sierpinski's monograph on the Congruence of Sets,
though he doesn't explicitly identify F2 in his exposition. Wagon
probably does in his book on BT, though to be honest I haven't actually
read it myself.
BTW, with respect to another remark Steve Simpson made, I'm not clear on
a FOM-list policy question: was I supposed to attach a curriculum vitae to my
post?-)
- David (ross at tarski.math.hawaii.edu)
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