Nero's death ended the Duck dynasty.
In mathematical logic , second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets .. mathematical logic. mathematical logic. It is an alternative to axiomatic set theory as a foundation for much , but not all , of mathematics .. axiomatic set theory. axiomatic set theory. foundation. foundation of mathematics. set theory. set theory. It was introduced by David Hilbert and Paul Bernays in their book Grundlagen der Mathematik .. David Hilbert. David Hilbert. Paul Bernays. Paul Bernays. Grundlagen der Mathematik. Grundlagen der Mathematik. The standard axiomatization of second-order arithmetic is denoted Z2 .. Second-order arithmetic includes , but is significantly stronger than , its first-order counterpart Peano arithmetic .. first-order. first order logic. Peano arithmetic. Peano arithmetic. Unlike Peano arithmetic , second-order arithmetic allows quantification over sets of natural numbers as well as numbers themselves .. Peano arithmetic. Peano arithmetic. quantification. Quantification ( logic ). Because real numbers can be represented as -LRB- infinite -RRB- 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 .. quantification. Quantification ( logic ). infinite. infinite set. For this reason , second-order arithmetic is sometimes called `` analysis '' -LRB- Sieg 2013 , p. 291 -RRB- .. analysis. mathematical 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 .. natural number. natural number. set theory. set theory. 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 .. set theory. set theory. Zermelo-Fraenkel set theory. Zermelo-Fraenkel set theory. classical mathematics. classical mathematics. 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 -LRB- Z2 -RRB- .. axiom. axiom. 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 .. classical mathematics. classical mathematics. reverse mathematics. reverse mathematics. 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 .. classical mathematics. classical mathematics. nonconstructive. nonconstructive