Coding the Universe

Abstract

Axiomatic set theory is the concern of this book. More particularly, the authors prove results about the coding of models M, of Zermelo-Fraenkel set theory together with the Generalized Continuum Hypothesis by using a class 'forcing' construction. By this method they extend M to another model L[a] with the same properties. L[a] is Gödels universe of 'constructible' sets L, together with a set of integers a which code all the cardinality and cofinality structure of M. Some applications are also considered. Graduate students and research workers in set theory and logic will be especially interested by this account.

Simplest possible locally definable well-orders

“…This is achieved (for a somewhat smaller class of cardinals κ) in the next theorem. 3 Remember that, given an uncountable cardinal κ, a subset S of κ is fat stationary if for every club C ⊆ κ, the intersection C ∩ S contains closed subsets of ordinals of arbitrarily large order-types below κ. Given regular cardinals η < κ, we let S κ η denote the set of all limit ordinals less than κ of cofinality η.…”

“…In this section, we discuss almost disjoint coding forcing (see [3] and [12]) for uncountable cardinals κ that satisfy κ = κ <κ . Given such κ, this forcing technique will allow us to make an arbitrary subset of κ κ definable by a formula of low complexity in an upwards-absolute way.…”

Coding into 𝐾 by reasonable forcing

“…We then let P 4 consist of all pairs p = (l(p), r(p)), where l(p): n → 2 for some n < ω and r(p) is a finite subset of ω 1 …”

Internal consistency for embedding complexity

“…These result use either reverse Easton or forward Easton forcing methods. An example of the latter is due to Jensen [1], who showed that in L[0 # ] there is a real which is class-generic but not set-generic over L. All of these early results however only produce models of GCH.…”