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An Abstract Condensation Property


Law, David Richard (1994) An Abstract Condensation Property. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/khcx-zk98.


Let A = (A, ... ) be a relational structure. Say that A has condensation if there is an F : A^(ω) → A such that for every partial order P, it is forced by P that substructures of P which are closed under F are isomorphic to elements of the ground model. Condensation holds if every structure in V, the universe of sets, has condensation. This property, isolated by Woodin, captures part of the content of the condensation lemmas for L, K and other "L-like" models. We present a variety of results having to do with condensation in this abstract sense. Section 1 establishes the absoluteness of condensation and some of its consequences. In particular, we show that if condensation holds in M, then M ╞ GCH and there are no measurable cardinals or precipitous ideals in M. The results of this section are due to Woodin. Section 2 contains a proof that condensation implies ◊_κ(E) for κ regular and E κ stationary. This is the main result of this thesis. The argument provides a new proof of the key lemma giving GCH. Section 2 also contains some information about the relationship between condensation and strengthenings of diamond. Section 3 contains partial results having to do with forcing "Cond(A)", some further discussion of the relation between condensation and combinatorial principles which hold in L, and an argument that Cond(G) fails in V[G], where G is generic for the partial order adding ω_2 cohen subsets of ω_1.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Woodin, W. Hugh
Thesis Committee:
  • Unknown, Unknown
Defense Date:23 September 1993
Record Number:CaltechTHESIS:11212011-135758866
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:6736
Deposited On:21 Nov 2011 22:20
Last Modified:16 Apr 2021 23:17

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