Concept
Second-order arithmetic
Overview
In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics.
A precursor to second-order arithmetic that involves third-order parameters was introduced by David Hilbert and Paul Bernays in their book Grundlagen der Mathematik. The standard axiomatization of second-order arithmetic is denoted by Z2.
Second-order arithmetic includes, but is significantly stronger than, its first-order counterpart Peano arithmetic. Unlike Peano arithmetic, second-order arithmetic allows quantification over sets of natural numbers as well as numbers themselves. Because real numbers can be represented as (infinite) sets of natural numbers in well-known ways, and because second-order arithmetic allows quantification over such sets, it is possible to formalize the real numbers in second-order arithmetic. For this reason, second-order arithmetic is sometimes called "analysis".
Second-order arithmetic can also be seen as a weak version of set theory in which every element is either a natural number or a set of natural numbers. Although it is much weaker than Zermelo–Fraenkel set theory, second-order arithmetic can prove essentially all of the results of classical mathematics expressible in its language.
A subsystem of second-order arithmetic is a theory in the language of second-order arithmetic each axiom of which is a theorem of full second-order arithmetic (Z2). Such subsystems are essential to reverse mathematics, a research program investigating how much of classical mathematics can be derived in certain weak subsystems of varying strength. Much of core mathematics can be formalized in these weak subsystems, some of which are defined below. Reverse mathematics also clarifies the extent and manner in which classical mathematics is nonconstructive.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
In modern expositions of the axioms of geometry certain conditions are laid down which the relation of congruence between segments is to satisfy. It is supposed that we have a complete theory of points, straight lines, planes, and the order of points on planes--in fact, a complete theory of non-metrical geometry. We then enquire about congruence and lay down the set of conditions--or axioms as they are called--which…
Alfred North Whitehead, The Concept of Nature The Tarner Lectures Delivered in Trinity College, November 1919
States what Second-order arithmetic is.
Any such select set is called the set of postulates, or premises, from which the reasoning proceeds. The reasoning is nothing else than the exhibition of the whole pattern of general conditions involved in the pattern derived from the selected postulates.
Alfred North Whitehead, Science and the modern world
States what Second-order arithmetic is.
All types of series are capable of formal definition, and their properties can be deduced from the principles of symbolic logic by means of the Algebra of Relatives. The notion of a limit, which is fundamental in the greater part of higher mathematics, used to be defined by means of quantity, as a term to which the terms of some series approximate as nearly as we please. But nowadays the limit is defined quite…
Bertrand Russell, Mysticism and Logic and Other Essays
States what Second-order arithmetic is.
Philosophers who discussed Second-order arithmetic
Related ideas
- First-order logic
- Infinite set
- Integer
- Zermelo–Fraenkel set theory
- Foundations of mathematics
- Mathematical logic
- Rational number
- Real number
- Natural number
- Domain of discourse
- Prefix
- Peano axioms
- Recursion
- Individual
- Set theory
- Subset
- Topology
- Second-order logic
- 0
- Theory (mathematical logic)
- Variable (mathematics)
- Well-formed formula
- Quantifier (logic)
- Mathematical analysis
Place in the tradition
Part of First-order logic.