Forcing and the Independence of the Continuum Hypothesis from ZFC

Specialeforsvar ved Alex Hein Larsen

Titel:  Forcing and the Independence of the Continuum Hypothesis from ZFC

Abstract:  This thesis uses the forcing method to prove that the Continuum Hypothesis is independent of the ZFC Axioms. The forcing method takes a model M of ZFC, a p.o. set P in M and a P-generic filter G and uses it to construct an extended model M[G] of ZFC. The forcing relation can be used to make statements about truth in M[G] from within M, allowing us to force certain statements to hold in M[G], most notably CH and -CH.
The existence of such models prove the relative consistency of ZFC with CH and :CH, implying the desired independence claim.
To construct a model where -CH holds, we force with nite partialfunctions to construct a model where 2^w! w₂.
To construct a model where CH holds, we force with the countable support p.o. set for collapsing  to  to construct a model where 2^w =w₁.
Finally, we investigate various possible equalities involving cardinals when the General Continuum Hypothesis fails, and develop a method to make pretty much any finite number of such equalities hold simultaneously.

Vejleder:   Asger D. Tørnqvist
Censor:     Jesper Bengtson, ITU