show/hide this revision's text 2 changed "real numbers without the property of Baire" to "a *set of* real numbers..."

I have no idea who these alleged people are who are nervous about using full AC but not countable AC, but perhaps they (as you would put it) cannot fathom the following consequences of ZFC which are unprovable from ZF + countable choice:

-The existence of a nonmeasurable set of real numbers;

-The existence of a set of real numbers without the property of Baire;

-The Banach-Tarski paradox (that the unit ball in R^3 has a finite, pairwise-disjoint decomposition into subsets which can be reassembled, via isometries in R^3, into two identical copies of the original unit ball).
show/hide this revision's text 1

I have no idea who these alleged people are who are nervous about using full AC but not countable AC, but perhaps they (as you would put it) cannot fathom the following consequences of ZFC which are unprovable from ZF + countable choice:

-The existence of a nonmeasurable set of real numbers;

-The existence of real numbers without the property of Baire;

-The Banach-Tarski paradox (that the unit ball in R^3 has a finite, pairwise-disjoint decomposition into subsets which can be reassembled, via isometries in R^3, into two identical copies of the original unit ball).