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.
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 = { □, ■□, ■■□, ■■■□, ■■■■□, ■■■■■□, ..., ■n□, etc. }
Read more about this topic: Object Theory
Famous quotes containing the words genetic, axiomatic and/or method:
“Nature, we are starting to realize, is every bit as important as nurture. Genetic influences, brain chemistry, and neurological development contribute strongly to who we are as children and what we become as adults. For example, tendencies to excessive worrying or timidity, leadership qualities, risk taking, obedience to authority, all appear to have a constitutional aspect.”
—Stanley Turecki (20th century)
“It is ... axiomatic that we should all think of ourselves as being more sensitive than other people because, when we are insensitive in our dealings with others, we cannot be aware of it at the time: conscious insensitivity is a self-contradiction.”
—W.H. (Wystan Hugh)
“I know no method to secure the repeal of bad or obnoxious laws so effective as their stringent execution.”
—Ulysses S. Grant (1822–1885)