Intermediate models and Kinna–Wagner Principles

Kinna–Wagner Principles state that every set can be mapped into some fixed iterated power set of an ordinal, and we write <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper K sans-serif upper W sans-serif upper P"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">K</mml:mi> <mml:mi mathvariant="sans-serif">W</mml:mi> <mml:mi mathvariant="sans-serif">P</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {KWP}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to denote that there is some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi> α </mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which this holds. The Kinna–Wagner Conjecture, formulated by the first author [Bull. Symb. Log., arXiv:2006.04514], states that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a model of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper Z sans-serif upper F plus sans-serif upper K sans-serif upper W sans-serif upper P"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">Z</mml:mi> <mml:mi mathvariant="sans-serif">F</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">K</mml:mi> <mml:mi mathvariant="sans-serif">W</mml:mi> <mml:mi mathvariant="sans-serif">P</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {ZF}+\mathsf {KWP}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -generic filter, then whenever <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an intermediate model of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper Z sans-serif upper F"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">Z</mml:mi> <mml:mi mathvariant="sans-serif">F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {ZF}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , that is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V subset-of-or-equal-to upper W subset-of-or-equal-to upper V left-bracket upper G right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo> ⊆ </mml:mo> <mml:mi>W</mml:mi> <mml:mo> ⊆ </mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">[</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">V\subseteq W\subseteq V[G]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W equals upper V left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>=</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">W=V(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfies <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper K sans-serif upper W sans-serif upper P"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">K</mml:mi> <mml:mi mathvariant="sans-serif">W</mml:mi> <mml:mi mathvariant="sans-serif">P</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {KWP}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In this work we prove the conjecture and generalise it even further. We include a brief historical overview of Kinna–Wagner Principles and new results about Kinna–Wagner Principles in the multiverse of sets.