Reverse mathematics and ordinal exponentiation

Reverse mathematics, ordinal numbers, and the ACC

A Survey of the Reverse Mathematics of Ordinal Arithmetic

This article surveys theorems of reverse mathematics concerning the comparability, addition, multiplication and exponentiation of countable well orderings. In [13], Simpson points out that ATR0 is “strong enough to accommodate a good theory of countable ordinal numbers, encoded by countable well orderings.” This paper provides a substantial body of empirical evidence supporting Simpson’s claim....

Reverse-engineering Reverse Mathematics

An important open problem in Reverse Mathematics ([16, 25]) is the reduction of the first-order strength of the base theory from IΣ1 to I∆0 + exp. The system ERNA, a version of Nonstandard Analysis based on the system I∆0 + exp, provides a partial solution to this problem. Indeed, Weak König’s lemma and many of its equivalent formulations from Reverse Mathematics can be ‘pushed down’ into ERNA,...

Reverse Mathematics

In math we typically assume a set of axioms to prove a theorem. In reverse mathematics, the premise is reversed: we start with a theorem and try to determine the minimal axiomatic system required to prove the theorem (over a weak base system). This produces interesting results, as it can be shown that theorems from different fields of math such as group theory and analysis are in fact equivalen...

ERNA and Friedman's Reverse Mathematics

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 Kö...