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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." [7] Euclidean geometry: Under Hilbert's axiom system the primitive notions are point, line, plane, congruence, betweenness , and incidence.
This permitted several primitive terms used by Hilbert to become defined entities, reducing the number of primitive notions to two, point and order. [37] Many other axiomatic systems for Euclidean geometry have been proposed over the years. A comparison of many of these can be found in a 1927 monograph by Henry George Forder. [53]
Removing five axioms mentioning "plane" in an essential way, namely I.4–8, and modifying III.4 and IV.1 to omit mention of planes, yields an axiomatization of Euclidean plane geometry. Hilbert's axioms, unlike Tarski's axioms, do not constitute a first-order theory because the axioms V.1–2 cannot be expressed in first-order logic.
Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", [1] and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded. Informally, [2] Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded ...
The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. [1] The axiom of choice is equivalent to the statement that every partition has a transversal. [2]
[52] [53] [54] The notions of angle and distance become primitive concepts. [55] Tarski's axioms: Alfred Tarski (1902–1983) and his students defined elementary Euclidean geometry as the geometry that can be expressed in first-order logic and does not depend on set theory for its logical basis, [56] in contrast to Hilbert's axioms, which ...
The primitive notions of his theory were function and argument. Using these notions, he defined class and set. [1] Paul Bernays reformulated von Neumann's theory by taking class and set as primitive notions. [2] Kurt Gödel simplified Bernays' theory for his relative consistency proof of the axiom of choice and the generalized continuum ...
In mathematics and logic, an axiomatic system is any set of primitive notions and axioms to logically derive theorems.A theory is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems.