User dr strangechoice - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T02:41:15Z http://mathoverflow.net/feeds/user/5702 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22927/why-worry-about-the-axiom-of-choice/22935#22935 Answer by Dr Strangechoice for Why worry about the axiom of choice? Dr Strangechoice 2010-04-29T04:32:41Z 2010-04-29T04:32:41Z <p><strong>How I Learned to Stop Worrying and Love the Axiom of Choice</strong></p> <p>The universe can be very a strange place without choice. One consequence of the Axiom of Choice is that when you partition a set into disjoint nonempty parts, then the number of parts does not exceed the number of elements of the set being partitioned. This can fail without the Axiom of Choice. In fact, if all sets of reals are Lebesgue measurable, then it is possible to partition $2^{\omega}$ into <em>more</em> than $2^{\omega}$ many pairwise disjoint nonempty sets!</p> http://mathoverflow.net/questions/22927/why-worry-about-the-axiom-of-choice/22935#22935 Comment by Dr Strangechoice Dr Strangechoice 2010-04-29T05:07:00Z 2010-04-29T05:07:00Z If every set of reals is Lebesgue measurable then $\omega_1 \nleq 2^{\omega}$, but then you can partition $2^{\omega}$, or rather $\mathcal{P}(\omega\times\omega)$, into $\aleph_1+2^{\omega} &gt; 2^{\omega}$ pieces by putting two wellorderings of $\omega$ in the same piece iff they have the same order type, and all non-wellorderings into singleton pieces.