Numbers less than or equal to 0 (such as −1) are not natural numbers (rather Integers). The publisher has graciously consented to permit diffusion of this introduction to NFU via the web. Elementary Set Theory with a Universal Set. In this text, S refers to the Peano axioms. J., and Edward Zalta, 2004, "Frege, Boolos, and Logical Objects," Journal of Philosophical Logic 33: 1–26.
#NATURAL NUMBERS HOW TO#
Anderson and Edward Zalta (2004) show how to repair it. The Russell paradox proved this system inconsistent, but George Boolos (1998) and David J. Hatcher (1982) derives Peano's axioms from several foundational systems, including ZFC and category theory, and from the system of Frege's Grundgesetze der Arithmetik using modern notation and natural deduction. Theorem: the natural numbers satisfy Peano’s axioms The definition of a finite set is given independently of natural numbers: ĭefinition : A set is finite if and only if any non empty family of its subsets has a minimal element for the inclusion order.ĭefinition: a cardinal n is a natural number if and only if there exists a finite set of which the cardinal is n.ĭefinition: the successor of a cardinal K is the cardinal K + 1 Īxiom of cardinality: The sets A and B are equinumerous if and only if Card(A) = Card(B)ĭefinition: the sum of cardinals K and L such as K= Card(A) and L = Card(B) where the sets A and B are disjoint, is Card (A ∪ B). The simplest way to introduce cardinals is to add a primitive notion, Card(), and an axiom of cardinality to ZF set theory (without axiom of choice). However, it does not work in the axiomatic set theory ZFC nor in certain related systems, because in such systems the equivalence classes under equinumerosity are proper classes rather than sets.įor enabling natural numbers to form a set, equinumerous classes are replaced by special sets, named cardinal. This definition works in type theory, and in set theories that grew out of type theory, such as New Foundations and related systems. This definition may appear circular, but it is not, because equinumerosity can be defined in alternate ways, for instance by saying that two sets are equinumerous if they can be put into one-to-one correspondence-this is sometimes known as Hume's principle. More formally, a natural number is an equivalence class of finite sets under the equivalence relation of equinumerosity. Gottlob Frege and Bertrand Russell each proposed defining a natural number n as the collection of all sets with n elements. Ravven and Quine refer to these sets as "counter sets". The set N and its elements, when constructed this way, are an initial part of the von Neumann ordinals. The existence of the set N is equivalent to the axiom of infinity in ZF set theory. The structure ⟨ N, 0, S⟩ is a model of the Peano axioms ( Goldrei 1996). In Zermelo–Fraenkel (ZF) set theory, the natural numbers are defined recursively by letting 0 =.
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