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## clique idea – Reference for “$mathrm{PFA}$ implies $L(mathbb{R}) cap bigcup_{1 leq ok

The preprint of the latest end result of Aspero and Schindler, “Martin’s Maximum$^{++}$ implies Woodin’s Axiom $(*)$“, mentions productive pointclasses, and states that “$mathrm{PFA}$ implies $L(mathbb{R}) cap bigcup_{1 leq k < omega} mathcal{P}(mathbb{R}^k)$ is productive” is a corollary of Steel’s paper “$mathrm{PFA}$ implies $mathrm{AD}^{L(mathbb{R})}$“, through “pattern inner model theoretic arguments”. For console of reference I give the definition of a productive pointclass under.

A pointclass $Gamma subset bigcup_{1 leq ok < omega} mathcal{P}(mathbb{R}^ok)$ is claimed to breathe *productive* iff

- $Gamma subset Gamma^{infty}$, the place $Gamma^{infty}$ is the pointclass consisting of all universally Baire units of reals,
- $Gamma$ is closed below enhances, i.e. for all $ok < omega$, if $D in Gamma cap mathcal{P}(mathbb{R}^{ok+1})$, then $mathbb{R}^{ok+1} setminus D in Gamma$,
- $Gamma$ is closed below projections, i.e. for all $ok < omega$, if $D in Gamma cap mathcal{P}(mathbb{R}^{ok+2})$, then

commence{equation*}

exists^{mathbb{R}} D := {vec{x} in mathbb{R}^{ok+1} : exists y (vec{x}^{glower}(y) in D)} in Gamma,

stop{equation*}

and - the closure of $Gamma$ below projections is preserved by clique forcing notions in a tremendous route: for all $ok < omega$, if $D in Gamma cap mathcal{P}(mathbb{R}^{ok+2})$, then

commence{equation*}

(exists^{mathbb{R}} D)^* = {vec{x} in mathbb{R}^{ok+1} : exists y (vec{x}^{glower}(y) in D^*)}

stop{equation*}

in all common extensions of $V$.

Now, I’m inquisitive concerning the following:

- Where did this judgement of productive pointclasses first strategy about?
- Is there a extra easy path to cite the result “$mathrm{PFA}$ implies $L(mathbb{R}) cap bigcup_{1 leq k < omega} mathcal{P}(mathbb{R}^k)$ is productive”?

Any references with reference to these two questions are very mighty appreciated.

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