Axiomatic system

In mathematics and logic, an axiomatic system or axiom system is a standard type of deductive logical structure, used also in theoretical computer science. It consists of a set of formal statements known as axioms that are used for the logical deduction of other statements. In mathematics these logical consequences of the axioms may be known as lemmas or theorems. A mathematical theory is an expression used to refer to an axiomatic system and all its derived theorems.

A proof within an axiomatic system is a sequence of deductive steps that establishes a new statement as a consequence of the axioms. By itself, the system of axioms is, intentionally, a syntactic construct: when axioms are expressed in natural language, which is normal in books and technical papers, the nouns are intended as placeholder words. The use of an axiomatic approach is a move away from informal reasoning, in which nouns may carry real-world semantic values, and towards formal proof. In a fully formal setting, a logical system such as predicate calculus must be used in the proofs. The contemporary application of formal axiomatic reasoning differs from traditional methods both in the exclusion of semantic considerations, and in the specification of the system of logic in use.

The axiomatic method in mathematics

The reduction of a body of propositions to a particular collection of axioms underlies mathematical research. This dependence was very prominent, and contentious, in the mathematics of the first half of the twentieth century, a period to which some major landmarks of the axiomatic method belong. The probability axioms of Andrey Kolmogorov, from 1933, are a salient example.[1] The approach was sometimes attacked at "formalism", because it cut away parts of the working intuitions of mathematicians, and those applying mathematics. In historical context, this alleged formalism is now discussed as deductivism, still a widespread philosophical approach to mathematics.[2]

David Hilbert "was the first who explicitly adopted the axiomatic method as an investigative framework for the study of the foundations of mathematics".[3] For Hilbert, a major foundational issue was the logical status of Cantor's set theory. To avoid infinite regress, primitive notions (axioms) should be stated in a system of definitions and propositions, in such a way such that each new term could be formally eliminated, by falling back on terms already introduced.[4]

Part of L. E. J. Brouwer's critique of Hilbert's entire program resulted in an axiomatisation of intuitionistic propositional logic by Arend Heyting.[5] It allowed constructivism in mathematics to be reconciled with "deductivism", by an exchange of logical calculus, under the title of the Brouwer–Heyting–Kolmogorov interpretation.

Timeline of axiomatic systems

Further information: History of mathematics

Major axiomatic systems were developed in the nineteenth century. They included non-Euclidean geometry, the foundations of real analysis, Georg Cantor's abstract set theory, Gottlob Frege's work on foundations, and Hilbert's applications of axiomatic method as a research tool.

Date Author Work Comments
Fourth century BCE to third century BCE Euclid of Alexandria The Elements Known as the earliest extant axiomatic presentation of Euclidean plane geometry, covering also parts of number theory.[6]
1829 Nikolai Lobachevsky О началах геометрии ("On the Origin of Geometry") Lobachevsky's paper is now recognised as the first publication on axiomatic plane geometry developed without the parallel axiom of Euclid, so founding the subject of non-Euclidean geometry.
1882/3 to 1890s Walther von Dyck Axioms for abstract group theory Von Dyck is credited with the now-standard group theory axioms.[7] It is clear from von Dyck's introduction of free groups that he was working with the standard concept of abstract group. It is not, however, evident whether the existence of inverse elements was axiomatic: it would follow from the semantic assumption that groups were permutation groups (permutations being invertible by definition) or geometric transformations with the same property. The discursive style of the period did not labour such points. James Pierpont, one of the American "postulate theorists", did have by 1896 a set of axioms for groups. It is of the modern type, though uniqueness of the identity element (for example) was not assumed.[8]
1889 Giuseppe Peano Arithmetices principia, nova methodo exposita After some earlier work of others, the Peano axioms provided an axiomatic basis for the arithmetical operations on natural numbers, and mathematical induction, that gained wide acceptance.
1898 Alfred North Whitehead Treatise on Universal Algebra Whitehead gave the first axiomatic system for Boolean algebra, as introduced by George Boole in fundamental work on logic and probability.[9]
1899 David Hilbert Grundlagen der Geometrie Presented what are now known as Hilbert's axioms, a revised axiomatization of solid geometry.
1908 to 1922 Ernst Zermelo and Abraham Fraenkel Zermelo-Fraenkel set theory Building on Zermelo set theory from 1908, the Zermelo-Fraenkel (ZF) theory provided an axiomatic basis for set theory with a clarified axiom system (adopting a restriction to first-order logic). With the addition of the axiom of choice, the ZFC theory provided a working foundation for much of classical mathematics..[10]
1911 to 1913 Alfred North Whitehead and Bertrand Russell Principia Mathematica (3 vols.) A work devoted to the principle of axiomatic formalization of mathematics, that addressed the set theory paradoxes by an idiosyncratic version of type theory.
1932 Oswald Veblen and J. H. C. Whitehead The Foundations of Differential Geometry (1932) The work gave the accepted axiomatic definition of differential manifold,[11] apart from certain issues with separation axioms.

Axiomatization

Axiomatization and proof

In mathematics, axiomatization is the process of taking a body of knowledge and working backwards towards its axioms. It is the formulation of a system of statements (i.e. axioms) that relate a number of primitive terms — in order that a consistent body of propositions may be derived deductively from these statements. Thereafter, the proof of any proposition should be, in principle, traceable back to these axioms.

If the formal system is not complete not every proof can be traced back to the axioms of the system it belongs. For example, a number-theoretic statement might be expressible in the language of arithmetic (i.e. the language of the Peano axioms) and a proof might be given that appeals to topology or complex analysis. It might not be immediately clear whether another proof can be found that derives itself solely from the Peano axioms.

Properties

Four important properties of an axiom system are consistency, relative consistency, completeness and independence. An axiomatic system is said to be consistent if it lacks contradiction. That is, it is impossible to derive both a statement and its negation from the system's axioms.[12] Consistency is a key requirement for most axiomatic systems, as the presence of contradiction would allow any statement to be proven (principle of explosion). Relative consistency comes into play when we can not prove the consistency of an axiom system. However, in some cases we can show that an axiom system A is consistent if another axiom set B is consistent.[12]

In an axiomatic system, an axiom is called independent if it cannot be proven or disproven from other axioms in the system. A system is called independent if each of its underlying axioms is independent.[12] Unlike consistency, in many cases independence is not a necessary requirement for a functioning axiomatic system — though it is usually sought after to minimize the number of axioms in the system.

An axiomatic system is called complete if for every statement, either itself or its negation is derivable from the system's axioms, i.e. every statement can be proven true or false by using the axioms.[12][13] However, note that in some cases it may be undecidable if a statement can be proven or not.

Axioms and models

A model for an axiomatic system is a formal structure, which assigns meaning for the undefined terms presented in the system, in a manner that is correct with the relations defined in the system. If an axiom system has a model, the axioms are said to have been satisfied.[14] The existence of a model which satisfies an axiom system, proves the consistency of the system.[15]

Models can also be used to show the independence of an axiom in the system. By constructing a model for a subsystem (without a specific axiom) shows that the omitted axiom is independent if its correctness does not necessarily follow from the subsystem.[14]

Two models are said to be isomorphic if a one-to-one correspondence can be found between their elements, in a manner that preserves their relationship.[16] An axiomatic system for which every model is isomorphic to another is called categorical or categorial. However, this term should not be confused with the topic of category theory. The property of categoriality (categoricity) ensures the completeness of a system, however the converse is not true: Completeness does not ensure the categoriality (categoricity) of a system, since two models can differ in properties that cannot be expressed by the semantics of the system.

See also

Wikiquote has quotations related to Axiomatic system.
  • Axiom schema – Short notation for a set of statements that are taken to be true
  • Formal system – Mathematical model for deduction or proof systems
  • Gödel's incompleteness theorems – Limitative results in mathematical logic
  • Hilbert-style deduction system – System of formal deduction in logic
  • History of logic
  • List of logic systems
  • Logicism – School of thought in philosophy of mathematics
  • Zermelo–Fraenkel set theory – Standard system of axiomatic set theory, an axiomatic system for set theory and today's most common foundation for mathematics.

References

  1. ^ Gauch, Hugh G. (2012). Scientific Method in Brief. Cambridge University Press. p. 136. ISBN 978-1-107-01962-1.
  2. ^ Paseau, Alexander; Pregel, Fabian (Aug 25, 2023). "Deductivism in the Philosophy of Mathematics". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
  3. ^ Baldwin, Thomas (27 November 2003). The Cambridge History of Philosophy 1870-1945. Cambridge University Press. p. 142. ISBN 978-0-521-59104-1.
  4. ^ Potter, Michael (15 January 2004). Set Theory and its Philosophy: A Critical Introduction. Clarendon Press. p. 6. ISBN 978-0-19-155643-2.
  5. ^ Baldwin, Thomas (27 November 2003). The Cambridge History of Philosophy 1870-1945. Cambridge University Press. p. 589. ISBN 978-0-521-59104-1.
  6. ^ Lehman, Eric; Meyer, Albert R; Leighton, F Tom. Mathematics for Computer Science (PDF). Retrieved 2 May 2023.
  7. ^ Kline, Morris (1 March 1990). Mathematical Thought From Ancient to Modern Times, Volume 3. Oxford University Press. p. 1141. ISBN 978-0-19-977048-9.
  8. ^ Zitarelli, David E.; Dumbaugh, Della; Kennedy, Stephen F. (28 July 2022). A History of Mathematics in the United States and Canada: Volume 2: 1900–1941. American Mathematical Society. p. 72. ISBN 978-1-4704-6730-2.
  9. ^ Padmanabhan, Ranganathan; Rudeanu, Sergiu (2008). Axioms for Lattices and Boolean Algebras. World Scientific. p. 73. ISBN 978-981-283-454-6.
  10. ^ Weisstein, Eric W. "Zermelo-Fraenkel Axioms". mathworld.wolfram.com. Retrieved 2019-10-31.
  11. ^ O'Connor, John J.; Robertson, Edmund F., "John Henry Constantine Whitehead", MacTutor History of Mathematics Archive, University of St Andrews
  12. ^ a b c d A. G. Howson A Handbook of Terms Used in Algebra and Analysis, Cambridge UP, ISBN 0521084342 1972 pp 6
  13. ^ Weisstein, Eric W. "Complete Axiomatic Theory". mathworld.wolfram.com. Retrieved 2019-10-31.
  14. ^ a b C. C. Chang and H. J. Keisler "Model Theory" Elsevier 1990, pp 1-7
  15. ^ C. C. Chang and H. J. Keisler "Model Theory" Elsevier 1990, pp 1-7, Theorem 1.2.11
  16. ^ Hodges, Wilfrid; Scanlon, Thomas (2018), "First-order Model Theory", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Winter 2018 ed.), Metaphysics Research Lab, Stanford University, retrieved 2019-10-31

Further reading

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