Second-order arithmetic: Difference between revisions

Second-order arithmetic, denoted \(Z_2\) is an extension of first-order (i.e. Peano) arithmetic by adding additional second-order variables as well as an induction scheme for \(\mathcal{P}(\mathbb{N})\), and a comprehension scheme. Proof-theoretically, \(Z_2\) is a very expressive system, as it can prove the consistency of Peano arithmetic and its extensions via the addition of iterated inductive definitions - an ordinal analysis of \(Z_2\) is considered the holy grail of ordinal analysis, and many believe it can be done using [[Bashicu matrix system|BMS]].

 

One of the primary interests regarding \(Z_2\) is the study of its subsystems, rather than the whole. This is part of a program called reverse mathematics. Since rational numbers, real numbers, complex numbers, continuous functions on the reals, countable groups, and more can be defined in the language of second-order arithmetic, it turns out many classical theorems in number theory, real analysis, topology, abstract algebra and group theory are provable in \(Z_2\), and most even in weak subsystems! The "big five" are the following:<ref>Subsystems of Second Order Arithmetic, Simpson, S.G., ''Perspectives in Logic'', 2009, ''Cambridge University Press''</ref>