| Summary: | I begin the study of a hierarchy of (hereditarily) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo><</mo><mi>κ</mi></mrow></semantics></math></inline-formula>-blurrily ordinal definable sets. Here for a cardinal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>κ</mi></semantics></math></inline-formula>, a set is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo><</mo><mi>κ</mi></mrow></semantics></math></inline-formula>-blurrily ordinal definable if it belongs to an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">OD</mi></semantics></math></inline-formula> set of cardinality less than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>κ</mi></semantics></math></inline-formula>, and it is hereditarily so if it and each member of its transitive closure is. I show that the class of hereditarily <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo><</mo><mi>κ</mi></mrow></semantics></math></inline-formula>-blurrily ordinal definable sets is an inner model of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">ZF</mi></semantics></math></inline-formula>. It satisfies the axiom of choice iff it is a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>κ</mi></semantics></math></inline-formula>-c.c. forcing extension of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">HOD</mi></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">HOD</mi></semantics></math></inline-formula> is definable inside it (even if it fails to satisfy the axiom of choice). Of particular interest are cardinals <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> such that some set is hereditarily <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo><</mo><mi>λ</mi></mrow></semantics></math></inline-formula>-blurrily ordinal definable but not hereditarily <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo><</mo><mi>κ</mi></mrow></semantics></math></inline-formula>-blurrily ordinal definable for any cardinal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>κ</mi><mo><</mo><mi>λ</mi></mrow></semantics></math></inline-formula>. Such cardinals I call leaps. The main results concern the structure of leaps. For example, I show that if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> is a limit of leaps, then the collection of all hereditarily <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo><</mo><mi>λ</mi></mrow></semantics></math></inline-formula>-blurrily ordinal definable sets is a model of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">ZF</mi></semantics></math></inline-formula> in which the axiom of choice fails. Using forcing, I produce models exhibiting various leap constellations, for example models in which there is a (regular/singular) limit leap whose cardinal successor is a leap. Many open questions remain.
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