Axiomatic system

An axiomatic system in mathematics is a set of axioms with rules of inference that allow theorems to be derived from the axioms.

Quotes

• A recurring concern has been whether set theory, which speaks of infinite sets, refers to an existing reality, and if so how does one ‘know’ which axioms to accept. It is here that the greatest disparity of opinion exists (and the greatest possibility of using different consistent axiom systems).
  • Paul Cohen: (2005). "Skolem and pessimism about proof in mathematics". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 363 (1835): 2407–2418. ISSN 1364-503X. DOI:10.1098/rsta.2005.1661. (quote from p. 2410)

• The idea that theorems follow from the postulates does not correspond to simple observation. If the Pythagorean theorem were found to not follow from the postulates, we would again search for a way to alter the postulates until it was true. Euclid's postulates came from the Pythagorean theorem, not the other way around.
  • Richard Hamming, "The Unreasonable Effectiveness of Mathematics", The American Mathematical Monthly 87 (2), February 1980, pp. 81-90

• The requisites for the axioms are various. They should be simple, in the sense that each axiom should enumerate one and only one statement. The total number of axioms should be few. A set of axioms must be consistent, that is to say, it must not be possible to deduce the contradictory of any axiom from the other axioms. According to the logical 'Law of Contradiction,' a set of entities cannot satisfy inconsistent axioms. Thus the existence theorem for a set of axioms proves their consistency. Seemingly this is the only possible method of proof of consistency.
  • Alfred North Whitehead: The axioms of projective geometry. Cambridge University Press. 1906. p. 3.