Asaf Karagila
I don't have much choice...

## Countable sets of reals

One of the classic results of Sierpinski is that if there are as many countable sets of reals as there are reals, then there is a set which is not Lebesgue measurable. (You can find a wonderful discussion on MathOverflow.)

This is fact is used in the paradoxical decomposition theorems (which I often enjoy bringing up as a counter-argument to bad arguments that the Banach–Tarski paradox implies we need to accept that all sets are measurable as an axiom):

## Cofinality and the axiom of choice

What is cofinality of a[n infinite] cardinal? If we think about the cardinals as ordinals, as we should in the case the axiom of choice holds, then the cofinality of a cardinal is just the smallest cardinality of an unbounded set. It can be thought of as the least ordinal from which there is an unbounded function into our cardinal. Or it could be thought as the smallest cardinality of a partition whose parts are all "small".

Not assuming the axiom of choice the definition of cofinality remains the same, if we restrict ourselves to ordinals and $$\aleph$$ numbers. But why should we? There is a rich world out there, new colors that were not on the choice-y rainbow from before. So anything which is inherently based on the ordering properties of the ordinals should not be considered as the definition of an ordinal. So first let's recall the two ways we can order cardinals without choice.