Local Hanf-Tarski numbers

It is shown that the least measurable cardinal is the Hanf-Tarski number of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L Subscript omega 1 comma omega"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi> ω </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mi> ω </mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {L}_{\omega _1,\omega }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <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> -logic, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L left-parenthesis upper Q 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>Q</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {L}(Q_1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Motivated by the distinction between a compactness number and a weak compactness number of a logic, we introduce the notion of a local Hanf-Tarski number of a logic and show that the least local Hanf-Tarski number of these logics is the least weakly compact cardinal. Indeed, we show that the local Hanf-Tarski numbers of these and other logics can be used to give new characterizations of various notions of largeness present in the literature and arising in varied contexts, including measurability, weak compactness, forms of unfoldability, and forms of weak extendibility, and subtlety. This is done via a number of “dichotomy” theorems whereby it is shown that the local Hanf-Tarski numbers of logics are either mildly large cardinals or limits of much larger cardinals. We also show that the assertion that every logic possesses a local Hanf-Tarski number is consistent with Gödel’s Axiom of Constructibility.