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Axiomatic foundations

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Axiomatic foundations of arithmetic try to provide a small set of laws, so-called axioms, 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 proofs in a rigorous manner. Two well-known approaches are the Dedekind–Peano axioms and set-theoretic constructions.[1][2][3]

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[a]. 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.[1][4][5]

  1. 0 is a natural number.
  2. For every natural number, there is a successor, which is also a natural number.
  3. The successors of two different natural numbers are never identical.
  4. 0 is not the successor of a natural number.
  5. If a set contains 0 and every successor then it contains every natural number.[1][6][5][7][b]

Numbers greater than 0 are expressed by repeated application of the successor function . For example, is and is . Arithmetic operations can be defined as mechanisms that affect how the successor function is applied. For instance, to add to any number is the same as applying the successor function two times to this number.[5][7]

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 . Each subsequent number can be defined as the union of the previous number with the set containing the previous number. For example, , , and .[8][9] Integers can be defined as ordered pairs of natural numbers where the second number is subtracted from the first one. For example, the pair (9, 0) represents the number 9 while the pair (0, 9) represents the number -9.[10][11] 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 .[12][11] One way to construct the real numbers relies on the concept of Dedekind cuts. According to this approach, each real number is represented by a partition of all rational numbers into two sets, one for all numbers below the represented real number and the other for the rest.[11][13] 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.[14]


  1. ^ The successor of a natural number is the number that comes after it. For instance, 4 is the successor of 3.
  2. ^ 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.


  • Bagaria, Joan (2023). "Set Theory". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Retrieved 19 November 2023.
  • Oliver, Alexander D. (2005). "arithmetic, foundations of". In Honderich, Ted (ed.). The Oxford Companion to Philosophy. Oxford University Press. ISBN 9780199264797.
  • Hosch, William L. (2010). "Peano axioms". Britannica. Retrieved 21 October 2023.
  • Tiles, Mary (8 July 2009). "A Kantian Perspective on the Philosophy of Mathematics". In Irvine, Andrew D. (ed.). Philosophy of Mathematics. Elsevier. ISBN 978-0-08-093058-9.
  • Ferreiros, Jose (22 November 2013). Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Birkhäuser. ISBN 978-3-0348-5049-0.
  • Ongley, John; Carey, Rosalind (17 January 2013). Russell: A Guide for the Perplexed. Bloomsbury Publishing. ISBN 978-1-4411-9123-6.
  • Taylor, Joseph L. (2012). Foundations of Analysis. American Mathematical Soc. ISBN 978-0-8218-8984-8.
  • Cunningham, Daniel W. (18 July 2016). Set Theory: A First Course. Cambridge University Press. ISBN 978-1-316-68204-3.
  • Hamilton, Norman T.; Landin, Joseph (16 May 2018). Set Theory: The Structure of Arithmetic. Courier Dover Publications. ISBN 978-0-486-83047-6.
  1. ^ a b c Oliver 2005, p. 58.
  2. ^ EoM staff 2020a.
  3. ^ Tiles 2009, A Kantian Perspective on the Philosophy of Mathematics.
  4. ^ Ferreiros 2013, p. 251.
  5. ^ a b c Ongley & Carey 2013, pp. 24–25.
  6. ^ Hosch 2010.
  7. ^ a b Taylor 2012, p. 8.
  8. ^ Bagaria 2023, § 3. The theory of transfinite ordinals and cardinals.
  9. ^ Cunningham 2016, pp. 83–84, 108.
  10. ^ Hamilton & Landin 2018, p. 133.
  11. ^ a b c Bagaria 2023, § 5. Set theory as the foundation of mathematics.
  12. ^ Hamilton & Landin 2018, pp. 157–158.
  13. ^ Hamilton & Landin 2018, p. 252.
  14. ^ Cunningham 2016, pp. 95–96.
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