Subsystems of Second Order Arithmetic

Abstract

Almost all of the problems studied in this book are motivated by an overriding foundational question: What are the appropriate axioms for mathematics? Through a series of case studies, these axioms are examined to prove particular theorems in core mathematical areas such as algebra, analysis, and topology, focusing on the language of second-order arithmetic, the weakest language rich enough to express and develop the bulk of mathematics. In many cases, if a mathematical theorem is proved from appropriately weak set existence axioms, then the axioms will be logically equivalent to the theorem. Furthermore, only a few specific set existence axioms arise repeatedly in this context, which in turn correspond to classical foundational programs. This is the theme of reverse mathematics, which dominates the first half of the book. The second part focuses on models of these and other subsystems of second-order arithmetic.

Reverse mathematics, trichotomy, and dichotomy

“…From reverse mathematics, we will concentrate on the axiom systems RCA 0 , WKL 0 , and ACA 0 , which are described in detail by Simpson [8]. Very roughly, RCA 0 is a subsystem of second order arithmetic incorporating ordered semi-ring axioms, a restricted form of induction, and comprehension for ∆ 0 1 definable sets.…”

“…Both Simpson [8] (following Definition II.4.4) and Kohlenbach [7] (in section 4.1) encode real numbers as rapidly converging sequences of rational numbers. In particular, Simpson defines a real number as a Cauchy sequence of rationals α = α(k) k∈N satisfying ∀k∀i(|α(k) − α(k + i)| ≤ 2 −k ).…”

10.4115/jla.v4i0.170

“…Proof Working in RCA 0 , item 2 is easily deduced from Σ 0 1 -separation, which is provable in WKL 0 as shown in Lemma IV.4.4 of Simpson [8].…”

“…By Lemma III.1.3 of Simpson [8], we need only show that item 3 suffices to prove the existence of ranges of injections. Let f : N → N be one-to-one.…”

10.4115/jla.v4i0.141

“…Extensive material related to the axiomatization and application of these subsystems to the study of complete separable metric spaces can be found in Simpson's book [5]. The codes for open sets and closed sets are sets of natural numbers, and thus elements of models of second order arithmetic.…”