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We study the influence of the existence of large cardinals on the existenceof wellorderings of power sets of infinite cardinals $\kappa$ with the propertythat the collection of all initial segments of the wellordering is definable bya $\Sigma_1$-formula with parameter $\kappa$. A short argument shows that theexistence of a measurable cardinal $\delta$ implies that such wellorderings donot exist at $\delta$-inaccessible cardinals of cofinality not equal to$\delta$ and their successors. In contrast, our main result shows that thesewellorderings exist at all other uncountable cardinals in the minimal modelcontaining a measurable cardinal. In addition, we show that measurability isthe smallest large cardinal property that interferes with the existence of suchwellorderings at uncountable cardinals and we generalize the above result tothe minimal model containing two measurable cardinals.

Measurable cardinals and good $\Sigma_1(\kappa)$-wellorderings