[FOM] New Axioms(?)
Bill Taylor
W.Taylor at math.canterbury.ac.nz
Thu Jun 26 23:37:23 EDT 2003
"Todd Eisworth (Math Fac)" <eisworth at math-cs.cns.uni.edu> writes:
->I am interested in people's opinions about how a "new axiom" gains
->acceptance.
As an out-of-the-blue opinion, I doubt that *any* new axioms will
"gain acceptance" any more, in any absolute terms. I think AC was
the last one, and lucky to make it at that. The nearest since would
be various large cardinal axioms, but so far folk have stuck rigidly
to the mantra "measurable cardinals imply that", or whatever. I do not
envisage this situation changing in my lifetime, nor any similar change
in set theory at least. (Other quite different FoMs may yet arrive with
their own "obvious" axioms, OC.)
->For a concrete question, what happened that makes the Axiom of Choice seem
->so much more reasonable to mathematicians now than 100 years ago?
No, it *did* seem quite reasonable then, just as it does now, to most mathies;
the only thing that's really changed since is that it is now in text-books
and undergraduate courses.
True, there *were* doubters back then, but there are also doubters now,
(including me).
->What does the Axiom of Choice possess that the Continuum Hypothesis does not?
Immediate applicability. There is hardly an area in C20 pure math that
doesn't involve it; and it makes for sweeping simplicity in a great many
areas. IMHO the turning point for unspoken acceptance came when some
of the loudest doubters (Borel etc) were found to have been using it
without observing so! But OC, they had only been using DC, which
very few folk doubt (me neither).
The only thing to perhaps have shaken a bit of absolute certainty about AC,
is AD; it strikes me that AD says *a very similar* sort of thing to AC,
namely that a certain sort of infinite choosing is allowable - but in
a much more restricted context, (integer sequences), so in some vague
Occamite sense it's "more likely" to be true; and it refutes full AC
of course. The fact that it *implies* DC is icing on the cake.
To those with a game-theoretic orientation, like me, it seems a much more
"likely" and "useful" axiom than full AC.
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Bill Taylor W.Taylor at math.canterbury.ac.nz
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Set theory is a shotgun marriage - between well-ordering and power-set.
The two parties get along OK; but they hardly seem made for each other.
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