High Quality Content by WIKIPEDIA articles! In mathematics, a shrewd cardinal is a certain kind of large cardinal number introduced by (Rathjen 1995)., extending the definition of indescribable cardinals. A cardinal number is called -shrewd if for every proposition , and set A V with (V + , , A) there exists an , ' < with (V + ', , A V ) . It is called shrewd if it is -shrewd for every (including > ). This definition extends the concept of indescribability to transfinite levels. A -shrewd cardinal is also -shrewd for any ordinal < . Shrewdness was developed by Michael Rathjen as part of his ordinal analysis of 12-comprehension. It is essentially the nonrecursive analog to the stability property for admissible ordinals. More generally, a cardinal number is called - m-shrewd if for every m proposition , and set A V with (V + , , A) there exists an , ' < with (V + ', , A V ) . Here one looks at formulas with m-1 alternations of quantifiers with the outermost quantifier being universal.

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