Baldwin: Axiomatic Set Theory and L_{omega_1,omega}

CUNY MODEL THEORY SEMINAR
Axiomatic Set Theory and $L_{\omega_1,\omega}$
John T. Baldwin (Mathematics, UIC)
Friday, April 20, 2012, 12:30 PM
Abstract. In the late 1960’s model theory and axiomatic set theory seemed to be inevitably intertwined. The fundamental notions of first order stability theory are absolute. We describe the role of this fact in the development of first order model theory independent from set theory since the 1970’s. The role of extensions of ZFC in infinitary logic is muddled. Important results are proved using weak extensions of ZFC; the use is not in general proved essential. We expound the following proof-scheme: (1) Prove an infinitary sentence is consistent with ZFC. (2) Prove there is a model of set theory for which this sentence is absolute. (3) Deduce the property it expresses is provable in ZFC. We will describe how this technique implies the following recent result: Theorem (Shelah) Let $\phi$ be a sentence of $L_{\omega_1, \omega}$. (a) If ‘algebraic closure’ fails exchange on models of phi then $\phi$ has many models in $\aleph_1$. (b) If $\phi$ is pseudo-minimal then it has model in the continuum. Here algebraic closure’ and pseudo-minimal’ are modifications of classical notions appropriate for the context.