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Measurable cardinals and good \(\Sigma_1(\kappa)\)-wellorderings. (English) Zbl 1521.03201

Summary: We study the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals \(\kappa\) with the property that the collection of all initial segments of the wellordering is definable by a \(\Sigma_1\)-formula with parameter \(\kappa \). A short argument shows that the existence of a measurable cardinal \(\delta\) implies that such wellorderings do not exist at \(\delta \)-inaccessible cardinals of cofinality not equal to \(\delta\) and their successors. In contrast, our main result shows that these wellorderings exist at all other uncountable cardinals in the minimal model containing a measurable cardinal. In addition, we show that measurability is the smallest large cardinal property that imposes restrictions on the existence of such wellorderings at uncountable cardinals. Finally, we generalise the above result to the minimal model containing two measurable cardinals.

MSC:

03E55 Large cardinals
03E35 Consistency and independence results
03E45 Inner models, including constructibility, ordinal definability, and core models
03E47 Other notions of set-theoretic definability
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