Introduction to symmetric models and the approximation lemma.

Last June, 2009, I gave a lecture on how to construct symmetric models at the Universiteit van Amsterdam. One student attended, Rogier Jacobs, and him I have to thank for attending and giving his input on the first version of the lecture notes. Here you can find the 'final' version of these notes:

Tutorial on symmetric models and the approximation lemma.

The notes assume basic knowledge of set theory and forcing, as given in Kunens "Set Theory", up to Chapter VII. In them it's shown how to construct symmetric models, and three basic examples are given. These are:

• the Feferman-Lévy model, where ω₁ is singular and the reals are a countable union of countable sets,
• a generalisation of the Jech model, where a successor cardinal is measurable, and
• a similar model to the Jech model, where a successor cardinal is weakly compact.

Soon there will be a follow-up note, with a small fragment of Gitik's construction in "All uncountable cardinals can be singular". In this small version of it, only one strongly compact cardinal is used (as opposed to the class of strongly compacts in Gitik's model), and the result is a symmetric model in which ω₁ is singular. Clearly this isn't an interesting result per se, but it will be presented as an introduction to the full Gitik construction. So there will be more notes appearing here, with the final goal of presenting Gitik's model where all uncountable cardinals are singular.

Comments on the pdf, and solved exercises for correction are very welcome!