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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." [9] Euclidean geometry: Under Hilbert's axiom system the primitive notions are point, line, plane, congruence, betweeness, and incidence.
Primitives (undefined terms) are the most basic ideas. Typically they include objects and relationships. In geometry, the objects are things like points, lines and planes while a fundamental relationship is that of incidence – of one object meeting or joining with another. The terms themselves are undefined.
Although these terms are not further defined, Euclid uses them to construct more complex geometric concepts. [5] Whether a particular function or value is undefined, depends on the rules of the formal system in which it is used. For example, the imaginary number is undefined within the set of real numbers.
defining the distance between two points P = (p x, p y) and Q = (q x, q y) is then known as the Euclidean metric, and other metrics define non-Euclidean geometries. In terms of analytic geometry, the restriction of classical geometry to compass and straightedge constructions means a restriction to first- and second-order equations, e.g., y = 2x ...
To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms. In other words, the elements of geometry form a system which is not susceptible of extension, if we regard the five groups of axioms as valid.
Created Date: 8/30/2012 4:52:52 PM
A good example is the relative consistency of absolute geometry with respect to the theory of the real number system. Lines and points are undefined terms (also called primitive notions) in absolute geometry, but assigned meanings in the theory of real numbers in a way that is consistent with both axiom systems. [citation needed]
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