A transfer principle for second-order arithmetic, and applications

Abstract

In the theory of conditional sets, many classical theorems from areas such as analysis, probability theory or measure theory are lifted to the conditional framework, often to be applied in areas such as mathematical economics or optimization. The frequent experience that such theorems can be proved by `conditionalizations' of the classical proofs suggests that a general transfer principle is in the background, and that formulating and proving such a transfer principle would yield a wealth of useful further conditional versions of classical results, in addition to providing a uniform approach to the results already known. In this paper, we formulate and prove such a transfer principle based on second-order arithmetic, which, by the results of reverse mathematics, suffices for the bulk of classical mathematics, including real analysis, measure theory and countable algebra, and excluding only more remote realms like category theory, set-theoretical topology or uncountable set theory, see e.g. the introduction of \cite{simpson2009subsystems}. This transfer principle is then employed to give short and easy proofs of conditional versions of central results in various areas of mathematics, including the Bolzano-Weierstrass theorem, the Heine-Borel theorem, the Riesz representation theorem and Brouwer's fixed point theorem.