FOM: Query on AC

Stephen G Simpson simpson at
Mon Sep 28 23:25:41 EDT 1998

Joe Shipman writes:
 > I wanted to see whether mathematically restricting AC to sets of
 > reals made it any more plausible.  

My feeling is that, on the contrary and perhaps somewhat
paradoxically, the full axiom of choice (= AC) is *more* plausible
than its restriction to sets of reals.  First, the naive idea of
choosing elements from nonempty sets seems completely general and
intuitive, almost a "law of thought"; to restrict it to sets of reals
strikes me as an artificial or arbitrary restriction.  Second, full AC
is equivalent to Zorn's lemma, which is traditionally regarded as a
nice principle with many nice consequences, while AC for sets of reals
leads to many pathological counterexamples, e.g. a non-measurable set,
or a set of reals X such that neither X nor R \ X contains a perfect

-- Steve

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