Pyramidal structure of constructibility degrees

Abstract

The degrees a0, 9 1 , . . . , and b could be obtained by means of the Sacks ~ x Sacks forcing. (Here Sacks ~ is the iteration of the Sacks forcing Sacks of length w with countable support; see [1]. Accordingly, Sacks m is the iteration of Sacks of length m.) However, in this process, another degree, the upper bound a~ of the degrees an , n E w, would emerge; this degree is incomparable with b, and so it is distinct from a~,b. Therefore, we need another form of iteration. Elements of the set w ~ will be called (real) numbers.