Category forcings, 𝑀𝑀⁺⁺⁺, and generic absoluteness for the theory of strong forcing axioms

Abstract

We analyze certain subfamilies of the category of complete boolean algebras with complete homomorphisms, families which are of particular interest in set theory. In particular we study the category whose objects are stationary set preserving, atomless complete boolean algebras and whose arrows are complete homomorphisms with a stationary set preserving quotient. We introduce a maximal forcing axiom <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="reverse-solidus textsf left-brace MM right-brace Superscript plus plus plus"> <mml:semantics> <mml:msup> <mml:mtext>\textsf {MM}</mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>+</mml:mo> <mml:mo>+</mml:mo> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\text {\textsf {MM}}^{+++}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as a combinatorial property of this category. This forcing axiom strengthens Martin’s maximum and can be seen at the same time as a strenghtening of Baire’s category theorem and of the axiom of choice. Our main results show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="reverse-solidus textsf left-brace MM right-brace Superscript plus plus plus"> <mml:semantics> <mml:msup> <mml:mtext>\textsf {MM}</mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>+</mml:mo> <mml:mo>+</mml:mo> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\text {\textsf {MM}}^{+++}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is consistent relative to large cardinal axioms and that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="reverse-solidus textsf left-brace MM right-brace Superscript plus plus plus"> <mml:semantics> <mml:msup> <mml:mtext>\textsf {MM}</mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>+</mml:mo> <mml:mo>+</mml:mo> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\text {\textsf {MM}}^{+++}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> makes the theory of the Chang model <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-parenthesis left-bracket reverse-solidus textrm left-brace Ord right-brace right-bracket Superscript less-than-or-equal-to normal alef 1 Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mtext>\textrm {Ord}</mml:mtext> <mml:msup> <mml:mo stretchy="false">]</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ≤ </mml:mo> <mml:msub> <mml:mi mathvariant="normal"> ℵ </mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L([\text {\textrm {Ord}}]^{\leq \aleph _1})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with parameters in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P left-parenthesis omega 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi> ω </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">P(\omega _1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> generically invariant for stationary set preserving forcings that preserve this axiom. We also show that our results give a close to optimal extension to the Chang model <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-parenthesis left-bracket reverse-solidus textrm left-brace Ord right-brace right-bracket Superscript less-than-or-equal-to normal alef 1 Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mtext>\textrm {Ord}</mml:mtext> <mml:msup> <mml:mo stretchy="false">]</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ≤ </mml:mo> <mml:msub> <mml:mi mathvariant="normal"> ℵ </mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L([\text {\textrm {Ord}}]^{\leq \aleph _1})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of Woodin’s generic absoluteness results for the Chang model <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-parenthesis left-bracket reverse-solidus textrm left-brace Ord right-brace right-bracket Superscript normal alef 0 Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mtext>\textrm {Ord}</mml:mtext> <mml:msup> <mml:mo stretchy="false">]</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal"> ℵ </mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L([\text {\textrm {Ord}}]^{\aleph _0})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and give an a posteriori explanation of the success forcing axioms have met in set theory.