Real analysis without classes

This paper explores strengths and limitations of both predicativism and nominalism, especially in connection with the problem of characterizing the continuum. Although the natural number structure can be recovered predicatively (despite appearances), no predicative system can characterize even the full predicative continuum which the classicist can recognize. It is shown, however, that the classical second-order theory of continua (third-order number theory) can be recovered nominalistically, by synthesizing mereology, plural quantification, and a modal-structured approach with essentially just the assumption that an ω-sequence of concrete atoms be possible. Predicative flexible type theory may then be used to carry out virtually all of scientifically applicable mathematics in a natural way, still without ultimate need of the platonist ontology of classes and relations.