Daisuke Ikegami (Shibaura Institute of Technology)

On preserving AD via forcings

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It is well-known that forcings preserve $\mathsf{ZFC}$, i.e., any set generic extension of any model of $\mathsf{ZFC}$ is again a model of $\mathsf{ZFC}$. How about the Axiom of Determinacy ($\mathsf{AD}$) under $\mathsf{ZF}$? It is not difficult to see that Cohen forcing always destroys $\mathsf{AD}$, i.e., any set generic extension of a model of $\mathsf{ZF}+ \mathsf{AD}$ via Cohen forcing is not a model of $\mathsf{AD}$. Actually it is open whether there is a forcing which adds a new real while preserving $\mathsf{AD}$. In this talk, we present some results on preservation & non-preservation of $\mathsf{AD}$ via forcings, whose details are as follows:

1. Starting with a model of $\mathsf{ZF} + \mathsf{AD}^{+} + V = L(P(R))$, any forcing increasing $\Theta$ destroys $\mathsf{AD}$.
2. It is consistent relative to $\mathsf{ZF} + \mathsf{AD}_R$ that $\mathsf{ZF} + \mathsf{AD}^{+} +$ There is a forcing which increases $\Theta$ while preserving $\mathsf{AD}$.
3. In $\mathsf{ZF}$, no forcings on the reals preserve $\mathsf{AD}$. (This is an improvement of the result of Chan and Jackson where they additionally assumed $\Theta$ is regular.)
4. In $\mathsf{ZF} + \mathsf{AD}^{+} + V = L(P(R)) + \Theta$ is regular, there is a forcing on $\Theta$ which adds a new subset of $\Theta$ while preserving $\mathsf{AD}$.

This is joint work with Nam Trang.