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Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. 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 which come into play. ==Axiomatic systems== (詳細は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. Hilbert once remarked that instead of points, lines and planes one might just as well talk of tables, chairs and beer mugs. His point being that the primitive terms are just empty shells, place holders if you will, and have no intrinsic properties. # Axioms (or postulates) are statements about these primitives; for example, ''any two points are together incident with just one line'' (i.e. that for any two points, there is just one line which passes through both of them). Axioms are assumed true, and not proven. They are the ''building blocks'' of geometric concepts, since they specify the properties that the primitives have. # The laws of logic. # The theorems〔In this context no distinction is made between different categories of theorems. Propositions, lemmas, corollaries, etc. are all treated the same.〕 are the logical consequences of the axioms, that is, the statements that can be obtained from the axioms by using the laws of deductive logic. An ''interpretation'' of an axiomatic system is some particular way of giving concrete meaning to the primitives of that system. If this association of meanings makes the axioms of the system true statements, then the interpretation is called a model of the system. In a model, all the theorems of the system are automatically true statements. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Foundations of geometry」の詳細全文を読む スポンサード リンク
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