Adrian Mathias (Université de la Réunion)

Power-admissible sets and ill-founded omega-models

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In the 1960s admissible sets were introduced which are transitive sets modelling principles of $\Sigma_1$ set-recursion.

In 1971 Harvey Friedman introduced power-admissible sets, which are transitive sets modelling principles of $\Sigma_1^P$,roughly$\Sigma_1$ recursion in the power-set function.

Several decades later I initiated the study of provident sets, which are transitive sets modelling principles of rudimentary recursion. Over the last fifty-odd years several workers have found that ill-founded omega-models, the axiom of constructibility and techniques from proof theory bring unexpected insights into the structure of these models of set-recursion.

In this talk I shall review these results and the methods of proof.