Measurable cardinals and good $Σ_1(κ)$-wellorderings

Philipp Lücke; Philipp Schlicht

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arXiv · 1704.00511

Measurable cardinals and good $Σ_1(κ)$-wellorderings

Philipp Lücke·Philipp Schlicht

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Abstract

We study the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals $κ$ with the property that the collection of all initial segments of the wellordering is definable by a $Σ_1$-formula with parameter $κ$. A short argument shows that the existence of a measurable cardinal $δ$ implies that such wellorderings do not exist at $δ$-inaccessible cardinals of cofinality not equal to $δ$ 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 interferes with the existence of such wellorderings at uncountable cardinals and we generalize the above result to the minimal model containing two measurable cardinals.

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BibTeX RIS

Philipp Lücke, Philipp Schlicht. 2017-04-03. Measurable cardinals and good $Σ_1(κ)$-wellorderings. https://arxiv.org/abs/1704.00511

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