7.2 The Hyperuniverse

In analysis for example real analysis or functional analysis, or in probability theory, it is not sufficient to consider alone, not even when all functions and subsets are added. We also need to consider higher objects such as probability measures and function spaces. These objects can be represented set-theoretically as sets of reals, sets of sets of reals, etc. So we would like to consider the accumulative hierarchy of sets built from by finite number of iterations of set operations. The result is called the superstructure over , denoted by V( ). It is remarkable that:

The entire ordinary mathematics can be formalized inside the superstructure V( ).

The practice of hyperanalysis involves extending V( ) to an enlarged structure * V( ), the hyperuniverse and proving theorems about V( ) by exploiting the relation between the two structures. The so-called internal sets in * V( ) satisfy principles analogous to those mention in 7.1, namely the principles of extension, transfer and saturation. Besides internal sets, two other kinds of objects are important: standard and external sets. Their properties will be carefully studied here.