Set theory of the reals

Set theory of the real line is an area of mathematics concerned with the application of set theory to aspects of the real numbers.

For example, one knows that all countable sets of reals are null, i.e. have Lebesgue measure 0; one might therefore ask the least possible size of a set which is not Lebesgue null. This invariant is called the uniformity of the ideal of null sets, denoted ${\displaystyle non({\mathcal {N}})}$. There are many such invariants associated with this and other ideals, e.g. the ideal of meagre sets, plus more which do not have a characterisation in terms of ideals. If the continuum hypothesis (CH) holds, then all such invariants are equal to ${\displaystyle \aleph _{1}}$, the least uncountable cardinal. For example, we know ${\displaystyle non({\mathcal {N}})}$ is uncountable, but being the size of some set of reals under CH it can be at most ${\displaystyle \aleph _{1}}$.

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