Oscillations of real numbers

Abstract

This chapter discusses sets of reals where the function osc takes all finite values. The first such set was found in connection with the Ramsey problem of the uncountable. It gave the first example of a partition of an uncountable square with properties very different from the properties of standard partitions of uncountable squares obtained by intersecting two total orderings. Because of the cardinal invariant involved, the example has a meager relevance to the Ramsey problem of the uncountable. However, it turned out to be useful in considering some other problems concerning the uncountable. The analysis of osc begins by introducing several useful technical definitions. The chapter also presents the main technical Lemma that shows how one finds two reals with a prescribed oscillation.