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An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (axíōma), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. [1] [2]
This is a list of axioms as that term is understood in mathematics. In epistemology , the word axiom is understood differently; see axiom and self-evidence . Individual axioms are almost always part of a larger axiomatic system .
The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view. The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry ...
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements.Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions from these.
Merely the use of formalism alone does not explain several issues: why we should use the axioms we do and not some others, why we should employ the logical rules we do and not some others, why "true" mathematical statements (e.g., the laws of arithmetic) appear to be true, and so on. Hermann Weyl posed these very questions to Hilbert:
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]
Old axiom II.5 (Pasch's Axiom) is renumbered as II.4. V.2, the Axiom of Line Completeness, replaced: Axiom of completeness. 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.
Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is geometry without the use of coordinates. It relies on the axiomatic method for proving all results from a few basic properties initially called postulates , and at present called axioms .