In case of Peano arithmetic or any familiar explicitly axiomatized theory T, it is possible to define the consistency "Con(T)" of T in terms of the non-existence of a number with a certain property, as follows: "there does not exist an integer coding a sequence of sentences, such that each sentence is either one of the (canonical) axioms of T, a logical axiom, or an immediate consequence of preceding sentences according to the rules of inference of first order logic, and such that the last sentence is a contradiction". However, for arbitrary T there is no canonical choice for Con(T).
In the mathematical field of group theory, the Monster group M or IM (also known as the Fischer-Griess Monster, or the Friendly Giant) is a group of finite order246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 = 808017424794512875886459904961710757005754368000000000 ˜ 8 · 1053.It is a simple group, meaning it does not have any normal subgroups except for the subgroup consisting only of the identity element, and M itself.The finite simple groups are believed to have been completely classified (the classification of finite simple groups), although some doubts remain as to whether the completeness of this classification has been completely proven. Assuming the classification is correct, the list of finite simple groups consists of 18 countably infinite families, plus 26 sporadic groups that do not follow such a systematic pattern. The Monster group is the largest of these sporadic groups and contains all but six of the other sporadic groups as subgroups or subquotients.
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The inverse or converse ,, of a partial order relation ,, satisfies x,,y if and only if y,,x. The inverse of a partial order relation is reflexive, transitive, and antisymmetric, and hence itself a partial order relation. The order dual of a partially ordered set is the same set with the partial order relation replaced by its inverse. The irreflexive relation is to ,, as on a given set uniquely determine the other three.In general two elements x and y of a partial order may stand in any of four mutually exclusive relationships to each other: either x y, or x and y are incomparable (none of the other three). A totally ordered set is one that rules out this fourth possibility: all pairs of elements are comparable and we then say that trichotomy holds. The natural numbers, the integers, the rationals, and the reals are all totally ordered by their algebraic (signed) magnitude whereas the complex numbers are not. This is not to say that the complex numbers cannot be totally ordered; we could for example order them lexicographically via x+iy u+iv if and only if x u or (x = u and y v), but this is not ordering by magnitude in any reasonable sense as it makes 1 greater than 100i. Ordering them by absolute magnitude yields a preorder in which all pairs are comparable, but this is not a partial order since 1 and i have the same absolute magnitude but are not equal, violating antisymmetry.
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