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We consider classical representations of integers: Church's functioniterators, cardinal equivalence classes of sets, ordinal equivalence classes oftotally ordered sets. Since programs do not work on abstract entities andrequire formal representations of objects, we effectivize these abstractnotions in order to allow them to be computed by programs. To any sucheffectivized representation is then associated a notion of Kolmogorovcomplexity. We prove that these Kolmogorov complexities form a strict hierarchywhich coincides with that obtained by relativization to jump oracles and/orallowance of infinite computations.

Church, Cardinal and Ordinal Representations of Integers and Kolmogorov complexity