Leonard Nimoy acted in a computer game.
In mathematics , the well-ordering principle states that every non-empty set of positive integers contains a least element .. mathematics. mathematics. least element. least element. integers. integers. In other words , the set of positive integers is well-ordered .. integers. integers. well-ordered. well-order. The phrase `` well-ordering principle '' is sometimes taken to be synonymous with the `` well-ordering theorem '' .. well-ordering theorem. well-ordering theorem. On other occasions it is understood to be the proposition that the set of integers -LCB- ... , − 2 , − 1 , 0 , 1 , 2 , 3 , ... -RCB- contains a well-ordered subset , called the natural numbers , in which every nonempty subset contains a least element .. least element. least element. integers. integers. well-ordered. well-order. natural numbers. natural numbers. Depending on the framework in which the natural numbers are introduced , this -LRB- second order -RRB- property of the set of natural numbers is either an axiom or a provable theorem .. natural numbers. natural numbers. axiom. axiom. For example  :. In Peano arithmetic , second-order arithmetic and related systems , and indeed in most -LRB- not necessarily formal -RRB- mathematical treatments of the well-ordering principle , the principle is derived from the principle of mathematical induction , which is itself taken as basic .. Peano arithmetic. Peano arithmetic. second-order arithmetic. second-order arithmetic. mathematical induction. mathematical induction. Considering the natural numbers as a subset of the real numbers , and assuming that we know already that the real numbers are complete -LRB- again , either as an axiom or a theorem about the real number system -RRB- , i.e. , every bounded -LRB- from below -RRB- set has an infimum , then also every set A of natural numbers has an infimum , say a * .. natural numbers. natural numbers. axiom. axiom. We can now find an integer n * such that a * lies in the half-open interval -LRB- n * − 1 , n * -RSB- , and can then show that we must have a * = n * , and n * in A.. In axiomatic set theory , the natural numbers are defined as the smallest inductive set -LRB- i.e. , set containing 0 and closed under the successor operation -RRB- .. natural numbers. natural numbers. axiomatic set theory. axiomatic set theory. inductive set. Inductive set ( axiom of infinity ). One can -LRB- even without invoking the regularity axiom -RRB- show that the set of all natural numbers n such that '' -LCB- 0 , ... , n -RCB- is well-ordered '' is inductive , and must therefore contain all natural numbers ; from this property one can conclude that the set of all natural numbers is also well-ordered .. well-ordered. well-order. natural numbers. natural numbers. axiom. axiom. regularity axiom. axiom of regularity. In the second sense , the phrase is used when that proposition is relied on for the purpose of justifying proofs that take the following form  : to prove that every natural number belongs to a specified set S , assume the contrary and infer the existence of a -LRB- non-zero -RRB- smallest counterexample .. Then show either that there must be a still smaller counterexample or that the smallest counterexample is not a counter example , producing a contradiction .. This mode of argument bears the same relation to proof by mathematical induction that `` If not B then not A '' -LRB- the style of modus tollens -RRB- bears to `` If A then B '' -LRB- the style of modus ponens -RRB- .. mathematical induction. mathematical induction. modus tollens. modus tollens. modus ponens. modus ponens. It is known light-heartedly as the `` minimal criminal '' method and is similar in its nature to Fermat 's method of `` infinite descent '' .. minimal criminal. minimal criminal. infinite descent. infinite descent. Garrett Birkhoff and Saunders Mac Lane wrote in A Survey of Modern Algebra that this property , like the least upper bound axiom for real numbers , is non-algebraic ; i.e. , it can not be deduced from the algebraic properties of the integers -LRB- which form an ordered integral domain -RRB- .. integers. integers. axiom. axiom. Garrett Birkhoff. Garrett Birkhoff. Saunders Mac Lane. Saunders Mac Lane. least upper bound axiom. least upper bound axiom. integral domain. integral domain