Combined maximality principles up to large cardinals.

*(English)*Zbl 1182.03078Let \(\varphi(a)\) 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.

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.

Reviewer: J. M. Plotkin (East Lansing)

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