This research is motivated by the program of Reverse Mathematics. We investigate basic part of complex analysis within some weak subsystems of second order arithmetic, in order to determine what kind of set existence axioms are needed to prove theorems of basic analysis. We are especially concerned with Cauchy’s integral theorem. We show that a weak version of Cauchy’s integral theorem is proved in RCAo. Using this, we can prove that holomorphic functions are analytic in RCAo. On the other hand, (...) we show that a full version of Cauchy’s integral theorem cannot be proved in RCAo but is equivalent to weak König’s lemma over RCAo. (shrink)

Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis with a PRA consistency proof, proposed around 1995 by Patrick Suppes and Richard Sommer. Recently, the author showed the consistency of ERNA with several transfer principles and proved results of nonstandard analysis in the resulting theories (see [12] and [13]). Here, we show that Weak König's lemma (WKL) and many of its equivalent formulations over RCA₀ from Reverse Mathematics (see [21] and [22]) can be 'pushed down' (...) into the weak theory ERNA, while preserving the equivalences, but at the price of replacing equality with equality 'up to infinitesimals'. It turns out that ERNA plays the role of RCA₀ and that transfer for universal formulas corresponds to WKL. (shrink)

In this paper, we introduce systems of nonstandard second-order arithmetic which are conservative extensions of systems of second-order arithmetic. Within these systems, we do reverse mathematics for nonstandard analysis, and we can import techniques of nonstandard analysis into analysis in weak systems of second-order arithmetic. Then, we apply nonstandard techniques to a version of Riemannʼs mapping theorem, and show several different versions of Riemannʼs mapping theorem.

Tanaka proved a powerful generalization of Friedman’s self-embedding theorem that states that given a countable nonstandard model \\) of the subsystem \ of second order arithmetic, and any element m of \, there is a self-embedding j of \\) onto a proper initial segment of itself such that j fixes every predecessor of m. Here we extend Tanaka’s work by establishing the following results for a countable nonstandard model \\ \)of \ and a proper cut \ of \:Theorem A. The (...) following conditions are equivalent: \ is closed under exponentiation. There is a self-embedding j of \\) onto a proper initial segment of itself such that I is the longest initial segment of fixed points of j.Theorem B. The following conditions are equivalent: \ is a strong cut of \ and \ There is a self-embedding j of \\) onto a proper initial segment of itself such that \ is the set of all fixed points of j. (shrink)