Studia Logica

, Volume 38, Issue 1, pp 17-36

First online:

Calculating self-referential statements, I: Explicit calculations

  • Craig SmorynskiAffiliated with

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The proof of the Second Incompleteness Theorem consists essentially of proving the uniqueness and explicit definability of the sentence asserting its own unprovability. This turns out to be a rather general phenomenon: Every instance of self-reference describable in the modal logic of the standard proof predicate obeys a similar uniqueness and explicit definability law. The efficient determination of the explicit definitions of formulae satisfying a given instance of self-reference reduces to a simple algebraic problem-that of solving the corresponding fixed-point equation in the modal logic. We survey techniques for the efficient calculation of such fixed-points.