The maximality of the core model

Abstract

Our main results are: 1) every countably certified extender that coheres with the core model <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is on the extender sequence of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, 2) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> computes successors of weakly compact cardinals correctly, 3) every model on the maximal 1-small construction is an iterate of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, 4) (joint with W. J. Mitchell) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K double-vertical-bar kappa"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mi>κ</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">K\|\kappa</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is universal for mice of height <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="less-than-or-equal-to kappa"> <mml:semantics> <mml:mrow> <mml:mo>≤</mml:mo> <mml:mi>κ</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\le \kappa</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whenever <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa greater-than-or-equal-to normal alef 2"> <mml:semantics> <mml:mrow> <mml:mi>κ</mml:mi> <mml:mo>≥</mml:mo> <mml:msub> <mml:mi mathvariant="normal">ℵ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\kappa \geq \aleph _2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, 5) if there is a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa"> <mml:semantics> <mml:mi>κ</mml:mi> <mml:annotation encoding="application/x-tex">\kappa</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa"> <mml:semantics> <mml:mi>κ</mml:mi> <mml:annotation encoding="application/x-tex">\kappa</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is either a singular countably closed cardinal or a weakly compact cardinal, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="white medium square Subscript kappa Superscript greater-than omega Baseline"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>◻</mml:mo> </mml:mrow> <mml:mi>κ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>&gt;</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">\square _\kappa ^{&gt;\omega }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> fails, then there are inner models with Woodin cardinals, and 6) an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi>ω</mml:mi> <mml:annotation encoding="application/x-tex">\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Erdös cardinal suffices to develop the basic theory of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.