Projective Determinacy

Part of the book series: Universitext ((UTX))

Abstract

We prove the Martin-Steel theorem according to which infinitely many Woodin cardinals give Projective Determinacy.

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1.
We here and in what follows use \(k {\dot{-}} l\) to denote \(k-l\), unless \(l>k\) in which case \(k {\dot{-}} l=0\).
2.
There is a slight abuse of notation here, as \({\fancyscript{T}}_s\) is not a set but rather a (sequence of) proper class(es). By \(({\fancyscript{T}}_s)^{N_{s',t'}}\) we mean that object which is defined over \(N_{s',t'}\) from the parameter \((E_{s,k}{:}\,k<2\cdot {\mathrm{lh}}(s))\) by the very same formula which defines \({\fancyscript{T}}_s\) over \(V\) from the same parameter \((E_{s,k} :k<2 \cdot 2 {\mathrm{lh}}(s))\).
3.
For a proper class \(X\), we write \(\sigma _{(\emptyset ,\emptyset ),(s',t')}(X)= \bigcup \{ \sigma _{(\emptyset ,\emptyset ),(s',t')}(X \cap V_\alpha ) :\alpha \in \mathrm{OR} \}\). Cf. the footnote on p. 184. As a matter of fact, in what follows we shall only need \(M_{s,k}^{s',t'}\) in case \(s\) and \(s'\) are compatible.

Institut für Mathematische Logik und Gru, Universität Münster, Münster, Germany
Ralf Schindler

Authors

Ralf Schindler
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Correspondence to Ralf Schindler .

Schindler, R. (2014). Projective Determinacy. In: Set Theory. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-06725-4_13