Set theory of the real line is an area of mathematics concerned with the application of set theory to aspects of the real numbers.
This invariant is called the uniformity of the ideal of null sets, denoted
If the continuum hypothesis (CH) holds, then all such invariants are equal to
is uncountable, but being the size of some set of reals under CH it can be at most
On the other hand, if one assumes Martin's Axiom (MA) all common invariants are "big", that is equal to
In fact one should view Martin's Axiom as a forcing axiom that negates the need to do specific forcings of a certain class (those satisfying the ccc, since the consistency of MA with large continuum is proved by doing all such forcings (up to a certain size shown to be sufficient).
Each invariant can be made large by some ccc forcing, thus each is big given MA.
If one restricts to specific forcings, some invariants will become big while others remain small.
Analysing these effects is the major work of the area, seeking to determine which inequalities between invariants are provable and which are inconsistent with ZFC.
Seventeen models (forcing constructions) were produced during the 1980s, starting with work of Arnold Miller, to demonstrate that no other inequalities are provable.
These are analysed in detail in the book by Tomek Bartoszynski and Haim Judah, two of the eminent workers in the field.
One of the last great unsolved problems of the area was the consistency of proved in 1998 by Saharon Shelah.