The Peirce translation

We develop applications of selection functions to proof theory and computational extraction of witnesses from proofs in classical analysis. The main novelty is a translation of minimal logic plus Peirce’s law into minimal logic, which we refer to as the Peirce translation, as it eliminates uses of Peirce’s law. When combined with modified realizability this translation applies to full classical analysis, i.e. Peano arithmetic in the language of finite types extended with countable choice and dependent choice. A fundamental step in the interpretation is the realizability of a strengthening of the double-negation shift via the iterated product of selection functions. In a separate paper we have shown that such a product of selection functions is equivalent, over system T, to modified bar recursion.