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Combined maximality principles up to large cardinals. (English) Zbl 1182.03078
Let $$\varphi(a)$$ be a formula and let $$\Gamma$$ be a definable (possibly with parameters) class of forcings. Then $$\varphi(a)$$ is said to be $$\Gamma$$-forceable if there is a $$\mathbb P\in \Gamma$$ such that $$\mathbb P\Vdash\varphi(\check{a})$$. Further, $$\varphi(a)$$ is $$\Gamma$$-necessary if for every $$\mathbb P\in \Gamma$$, $$\mathbb P\Vdash\varphi(\check{a})$$. It is $$\Gamma$$-forceably necessary if there is a $$\mathbb P\in\Gamma$$ such that $$\mathbb P\Vdash\forall\mathbb Q(\mathbb Q\in\Gamma\Rightarrow\mathbb Q\Vdash\varphi(\check{a}))$$. The Maximality Principle for forcings in $$\Gamma$$ with parameters in $$P$$, denoted by $$\text{MP}_{\Gamma}(P)$$, is the formula scheme that says that every formula with parameters in $$P$$ that is $$\Gamma$$-forceably necessary is true.
In the paper under review, the author investigates the possibility of combining Maximality Principles where the forcings involved are $$<\kappa$$-closed or $$<\kappa$$-directed-closed and $$P\subseteq H_{\kappa^+}$$, for $$\kappa$$ a regular cardinal. He begins by constructing models where the directed closed Maximality Principle holds below a large cardinal and shows that certain combinations have high consistency strength. He follows with the construction of models where the directed closed Maximality Principle holds up to and including a large cardinal. Finally, the author considers combinations of Maximality Principles up to and including cardinals that are partially supercompact or Woodinized supercompact.

##### MSC:
 3e+35 Consistency and independence results 3e+55 Large cardinals
##### Keywords:
maximality principle; large cardinal
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##### References:
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