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

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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

1. $$Gamma subset Gamma^{infty}$$, the place $$Gamma^{infty}$$ is the pointclass consisting of all universally Baire units of reals,
2. $$Gamma$$ is closed below enhances, i.e. for all $$ok < omega$$, if $$D in Gamma cap mathcal{P}(mathbb{R}^{ok+1})$$, then $$mathbb{R}^{ok+1} setminus D in Gamma$$,
3. $$Gamma$$ 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
4. 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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