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* [[Set theory]]: The concept of the [[Set (mathematics)|set]] is an example of a primitive notion. As [[Mary Tiles]] writes:<ref>[[Mary Tiles]] (2004) ''The Philosophy of Set Theory'', p. 99</ref> [The] 'definition' of 'set' is less a definition than an attempt at explication of something which is being given the status of a primitive, undefined, term. As evidence, she quotes [[Felix Hausdorff]]: "A set is formed by the grouping together of single objects into a whole. A set is a plurality thought of as a unit."
* [[Naive set theory]]: The [[empty set]] is a primitive notion. To assert that it exists would be an implicit [[axiom]].
* [[Peano arithmetic]]: The [[successor function]] and the number [[zero]] are primitive notions. Since Peano arithmetic is useful in regards to properties of the numbers, the objects that the primitive notions represent may not strictly matter.<ref>{{
* [[Axiomatic system]]s: The primitive notions will depend upon the set of axioms chosen for the system. [[Alessandro Padoa]] discussed this selection at the [[International Congress of Philosophy]] in Paris in 1900.<ref>[[Alessandro Padoa]] (1900) "Logical introduction to any deductive theory" in [[Jean van Heijenoort]] (1967) ''A Source Book in Mathematical Logic, 1879–1931'', [[Harvard University Press]] 118–23</ref> The notions themselves may not necessarily need to be stated; Susan Haack (1978) writes, "A set of axioms is sometimes said to give an implicit definition of its primitive terms."<ref>{{citation|first=Susan|last=Haack|year=1978|title=Philosophy of Logics|page=245|publisher=[[Cambridge University Press]]|isbn=9780521293297}}</ref>
* [[Euclidean geometry]]: Under [[Hilbert's axiom system]] the primitive notions are ''point, line, plane, congruence, betweeness'', and ''incidence''.
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