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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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