In category theory, a natural numbers object (NNO) is an object endowed with a recursive structure similar to natural numbers. More precisely, in a category...

mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative...

assignments of natural numbers to mathematical objects. Gödel noted that each statement within a system can be represented by a natural number (its Gödel...

Commonly encountered mathematical objects include numbers, expressions, shapes, functions, and sets. Mathematical objects can be very complex; for example...

adjectives, or adverbs that designate numbers. The distinction is drawn between the number five (an abstract object equal to 2+3), and the numeral five...

−2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set of all integers is often...

prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is...

rig of natural numbers N is an initial object. The zero rig, which is the zero ring, consisting only of a single element 0 = 1 is a terminal object. In Field...

mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented...

each object the size of the initial segment with that object as last element. Note that these numbers are one more than the formal ordinal numbers according...

The Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named...

objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers....

In set theory, several ways have been proposed to construct the natural numbers. These include the representation via von Neumann ordinals, commonly employed...

Object-oriented programming (OOP) is a programming paradigm based on the concept of objects. Objects can contain data (called fields, attributes or properties)...

of natural numbers. For example, the set of real numbers or the powerset of the set of natural numbers. Cardinal numbers extend the natural numbers as...

natural numbers using computable functions, to these different types of objects. A simple extension is to assign cardinal numbers to physical objects...

Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano...

In natural language and physical science, a physical object or material object (or simply an object or body) is a contiguous collection of matter, within...

Solar System planets and their natural satellites charts the progress of the discovery of new bodies over history. Each object is listed in chronological...

as a category, PER models are a cartesian closed category with natural numbers object and subobject classifier, giving rise to the effective topos introduced...

all square roots of natural numbers, other than of perfect squares, are irrational. Like all real numbers, irrational numbers can be expressed in positional...

mathematical object. For another axiomatization of R {\displaystyle \mathbb {R} } see Tarski's axiomatization of the reals. The real numbers can be constructed...

{\displaystyle {\mathsf {Sets}}\to {\mathsf {Eff}}} . The topos has a natural numbers object N = ⟨ N , E N ⟩ {\displaystyle N=\langle {\mathbb {N} },E_{\mathbb...

as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all...

construction of numbers and ordinal arithmetic, namely, the integers, reals, the countable infinity, and entire towers of infinite ordinals. The object to which...

when x and y are the same object. This is not the case for surreal number forms, but is true by construction for surreal numbers (equivalence classes). The...

or cardinality is therefore a natural number. For dealing with the case of infinite sets, the infinite cardinal numbers have been introduced, which are...

with the set of natural numbers. Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that...

formulations regarding finite objects tends to not differ from their classical counterparts. Given a model of all natural numbers, the equivalent for predicates...

referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers, and complex numbers. Addition belongs to arithmetic...