Master of Science (MS)
Maurice C. Stanley
Forcing, Set theory
If κ is an infinite cardinal, then a subset X of the reals R is called κ-dense if between any a < b, both in R, there are exactly κ elements of X. In these terms, a famous result of Cantor says that in every model of set theory all aleph-0-dense sets of reals are isomorphic (to the rationals, Q). This result cannot be directly extended, however, since for κ =aleph-1 there exist models of set theory in which not all aleph-1-dense sets are isomorphic. On the other hand, Baumgartner has shown by the method of iterated forcing that assuming the consistency of set theory, there does exist at least one model of set theory in which all aleph-1-dense subsets are isomorphic. We present here a detailed, yet expository, account of Baumgartner's result and discuss its relevance to the Proper Forcing Axiom of contemporary set theory.
Vartanian, Michael Haig, "An Iterated Forcing Extension In Which All Aleph-1 Dense Sets of Reals Are Isomorphic" (2010). Master's Theses. 3834.