Descriptive Set Theory - definable sets can be partitioned into more pieces than elements of original set

Question

I just read that "In Descriptive Set Theory .... definable sets can .... counter intuitively .... be partitioned into more pieces than there are elements in the original set".

Would anyone have a reference which discusses / proves it ?

Many Thanks

Answer 1

This sounds like an incomplete quote. The phenomenon being described is less about definable sets than definable maps.

For example, consider the Vitali relation on the reals: $$a\sim b\quad\iff\quad a-b\in\mathbb{Q}.$$ The resulting quotient set $\mathbb{R}_\sim:=\mathbb{R}/\sim$ (renamed for kerning purposes) is nicely definable. Moreover, with some care you can cook up a nicely-definable injection $\mathbb{R}\rightarrow \mathbb{R}_\sim$. However, in a precise sense there is no nicely-definable injection $\mathbb{R}_\sim\rightarrow \mathbb{R}$ (e.g. take "nicely-definable" to be "Borel"). So once we agree to compare sets only using "nicely definable" maps, we have a set which is smaller than one of its quotient sets.

Note that $\mathbb{R}_\sim$ and $\mathbb{R}$ "classically" have the same cardinality (assuming Choice of course - and in a sense, the definable-cardinality idea pointed out here is a "Choice-ification" of some combinatorics that can happen more literally in $\mathsf{ZF}$).