A transfer principle for second-order arithmetic, and applications

TL;DR

This paper develops a transfer principle based on second-order arithmetic that allows classical theorems in mathematics to be extended into a conditional framework, simplifying proofs and broadening applicability.

Contribution

It formulates and proves a general transfer principle for second-order arithmetic, enabling easier derivation of conditional versions of classical theorems across various mathematical fields.

Findings

Simplified proofs of conditional theorems like Peano existence and Urysohn's lemma.

Extension of classical results to conditional frameworks in functional analysis and measure theory.

Comparison of conditional models with standard models for key structures.

Abstract

In the theory of conditional sets, many classical theorems from areas such as functional analysis, probability theory or measure theory are lifted to a 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 theorems which have not been proven by hand previously such as Peano existence theorem, Urysohn's lemma and the Markov-Kakutani fixed point theorem. Moreover, we compare the interpretation of certain structures in a conditional model with their meaning in a standard model.