We show that each of the five basic theories of second order arithmetic that play a central role in reverse mathematics has a natural counterpart in the language of nonstandard arithmetic. In the earlier paper [3] we introduced saturation principles in nonstandard arithmetic which are equivalent in strength to strong choice axioms in second order arithmetic. This paper studies principles which are equivalent in strength to weaker theories in second order arithmetic.

This research is motivated by the program of reverse mathematics and non-standard arguments in second-order arithmetic. Within a weak subsystem of second-order arithmetic ACA0, we investigate some aspects of non-standard analysis related to sequential compactness. Then, using arguments of non-standard analysis, we show the equivalence of the Riemann mapping theorem and ACA0 over WKL0. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim).

In this paper, we show within ${\mathsf{RCA}_0}$ that both the Jordan curve theorem and the Schönflies theorem are equivalent to weak König’s lemma. Within ${\mathsf {WKL}_0}$ , we prove the Jordan curve theorem using an argument of non-standard analysis based on the fact that every countable non-standard model of ${\mathsf {WKL}_0}$ has a proper initial part that is isomorphic to itself (Tanaka in Math Logic Q 43:396–400, 1997).

Within a weak subsystem of second‐order arithmetic WKL0, we develop basic part of non‐standard analysis up to the Peano existence theorem.

We show that certain model-theoretic forcing arguments involving subsystems of second-order arithmetic can be formalized in the base theory, thereby converting them to effective proof-theoretic arguments. We use this method to sharpen the conservation theorems of Harrington and Brown-Simpson, giving an effective proof that WKL+0 is conservative over RCA0 with no significant increase in the lengths of proofs.