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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:
03E35 Consistency and independence results
03E55 Large cardinals
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