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Object theory is a theory in philosophy and mathematical logic concerning objects and the statements that can be made about objects. In some cases "objects" can be concretely thought of as symbols and strings of symbols, here illustrated by a string of four symbols " ←←↑↓←→←↓" as composed from the 4-symbol alphabet . When they are "known only through the relationships of the system (which they appear ), the system is (to be ) ''abstract'' ... what the objects are, in any respect other than how they fit into the structure, is left unspecified." (Kleene 1952:25) A further specification of the objects results in a model or representation of the abstract system, "i.e. a system of objects which satisfy the relationships of the abstract system and have some further status as well" (ibid). A system, in its general sense, is a collection of objects O = and (a specification of) the relationship ''r'' or relationships r1, r2, ... rn between the objects: : Example: Given a simple system = for a very simple relationship between the objects as signified by the symbol ∫ :〔Abstractly, the relationship ∫ is defined by the collection of ordered pairs 〕 :: ∫→ => ↑, ∫↑ => ←, ∫← => ↓, ∫↓ => → A model of this system would occur when we assign, for example the familiar natural numbers , to the symbols , i.e. in this manner: → = 0, ↑ = 1, ← = 2, ↓ = 3 . Here, the symbol ∫ indicates the "successor function" (often written as an apostrophe ' to distinguish it from +) operating on a collection of only 4 objects, thus 0' = 1, 1' = 2, 2' = 3, 3' = 0. :Or, we might specify that ∫ represents 90-degree counter-clockwise rotations of a simple object → . == The genetic versus axiomatic method == The following is an example of the genetic or constructive method of making objects in a system, the other being the axiomatic or postulational method. Kleene states that a genetic method is intended to "generate" all the objects of the system and thereby "determine the abstract structure of the system completely" and uniquely (and thus define the system categorically). If axioms rather than a genetic method is used, such axiom-sets are said to be categorical.〔Kleene 1952:26. This distinction between the constructive and axiomatic methods, and the words used to describe them, are Kleene's per his reference to Hilbert 1900.〕 Unlike the ∫ example above, the following creates an unbounded number of objects. The fact that O is a set, and □ is an element of O, and ■ is an operation, must be specified at the outset; this is being done in the language of the metatheory (see below): : Given the system ( O, □, ■ ): O = 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「Object theory」の詳細全文を読む スポンサード リンク
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