Editing Arithmetic (section)

== Axiomatic foundations ==
[[Axiomatic system|Axiomatic foundations]] of arithmetic try to provide a small set of laws, called [[axiom]]s, from which all fundamental properties of and operations on numbers can be derived. They constitute logically consistent and systematic frameworks that can be used to formulate [[mathematical proof]]s in a rigorous manner. Two well-known approaches are the [[Dedekind–Peano axioms]] and [[set-theoretic]] constructions.<ref>{{multiref | {{harvnb|Oliver|2005|p=58}} | {{harvnb|Bukhshtab|Pechaev|2020}} | {{harvnb|Tiles|2009|p=[https://books.google.com/books?id=mbn35b2ghgkC&pg=PA243 243]}} }}</ref>

The Dedekind–Peano axioms provide an axiomatization of the arithmetic of natural numbers. Their basic principles were first formulated by [[Richard Dedekind]] and later refined by [[Giuseppe Peano]]. They rely only on a small number of primitive mathematical concepts, such as 0, natural number, and [[Successor function|successor]].{{efn|The successor of a natural number is the number that comes after it. For instance, 4 is the successor of 3.}} The Peano axioms determine how these concepts are related to each other. All other arithmetic concepts can then be defined in terms of these primitive concepts.<ref>{{multiref | {{harvnb|Oliver|2005|p=58}} | {{harvnb|Ferreiros|2013|p=[https://books.google.com/books?id=-WL0BwAAQBAJ&pg=PA251 251]}} | {{harvnb|Ongley|Carey|2013|pp=[https://books.google.com/books?id=x-_KDwAAQBAJ&pg=PA26 26–27]}} }}</ref>

* 0 is a natural number.
* For every natural number, there is a successor, which is also a natural number.
* The successors of two different natural numbers are never identical.
* 0 is not the successor of a natural number.
* If a set contains 0 and every successor then it contains every natural number.<ref>{{multiref | {{harvnb|Oliver|2005|p=58}} | {{harvnb|Ongley|Carey|2013|pp=[https://books.google.com/books?id=x-_KDwAAQBAJ&pg=PA26 26–27]}} | {{harvnb|Xu|Zhang|2022|p=[https://books.google.com/books?id=s2RXEAAAQBAJ&pg=PA121 121]}} }}</ref>{{efn|There are different versions of the exact formulation and number of axioms. For example, some formulations start with 1 instead of 0 in the first axiom.<ref>{{harvnb|Taylor|2012|p=[https://books.google.com/books?id=0ky2W1loV-QC&pg=PA8 8]}}</ref>}}

Numbers greater than 0 are expressed by repeated application of the successor function <math>s</math>. For example, <math>1</math> is <math>s(0)</math> and <math>3</math> is <math>s(s(s(0)))</math>. Arithmetic operations can be defined as mechanisms that affect how the successor function is applied. For instance, to add <math>2</math> to any number is the same as applying the successor function two times to this number.<ref>{{multiref | {{harvnb|Ongley|Carey|2013|pp=[https://books.google.com/books?id=x-_KDwAAQBAJ&pg=PA26 26–27]}} | {{harvnb|Taylor|2012|p=[https://books.google.com/books?id=0ky2W1loV-QC&pg=PA8 8]}} }}</ref>

Various axiomatizations of arithmetic rely on set theory. They cover natural numbers but can also be extended to integers, rational numbers, and real numbers. Each natural number is represented by a unique set. 0 is usually defined as the empty set <math>\varnothing</math>. Each subsequent number can be defined as the [[Union (set theory)|union]] of the previous number with the set containing the previous number. For example, <math>1 = 0 \cup \{0\} = \{0\}</math>, <math>2 = 1 \cup \{1\} = \{0, 1\}</math>, and <math>3 = 2 \cup \{2\} = \{0, 1, 2\}</math>.<ref>{{multiref | {{harvnb|Bagaria|2023|loc=§ 3. The Theory of Transfinite Ordinals and Cardinals}} | {{harvnb|Cunningham|2016|pp=[https://books.google.com/books?id=fdigDAAAQBAJ&pg=PA83 83–84, 108]}} }}</ref> Integers can be defined as [[ordered pair]]s of natural numbers where the second number is subtracted from the first one. For instance, the pair (9, 0) represents the number 9 while the pair (0, 9) represents the number −9.<ref>{{multiref | {{harvnb|Hamilton|Landin|2018|p=[https://books.google.com/books?id=9aReDwAAQBAJ&pg=PA133 133]}} | {{harvnb|Bagaria|2023|loc=§ 5. Set Theory as the Foundation of Mathematics}} }}</ref> Rational numbers are defined as pairs of integers where the first number represents the numerator and the second number represents the denominator. For example, the pair (3, 7) represents the rational number <math>\tfrac{3}{7}</math>.<ref>{{multiref | {{harvnb|Hamilton|Landin|2018|pp=[https://books.google.com/books?id=9aReDwAAQBAJ&pg=PA157 157–158]}} | {{harvnb|Bagaria|2023|loc=§ 5. Set Theory as the Foundation of Mathematics}} }}</ref> One way to construct the real numbers relies on the concept of [[Dedekind cut]]s. According to this approach, each real number is represented by a [[Partition of a set|partition]] of all rational numbers into two sets, one for all numbers below the represented real number and the other for the rest.<ref>{{multiref | {{harvnb|Bagaria|2023|loc=§ 5. Set Theory as the Foundation of Mathematics}} | {{harvnb|Hamilton|Landin|2018|p=[https://books.google.com/books?id=9aReDwAAQBAJ&pg=PA252 252]}} }}</ref> Arithmetic operations are defined as functions that perform various set-theoretic transformations on the sets representing the input numbers to arrive at the set representing the result.<ref>{{harvnb|Cunningham|2016|pp=[https://books.google.com/books?id=fdigDAAAQBAJ&pg=PA95 95–96]}}</ref>