Well-ordered Set(Note: You will need Internet Explorer 6.0.2x or greater to view the mathematical symbols in this page correctly)
Coined by Georg Cantor as part of his theory of transfinite numbers, the mathematical pioneer's definition of well-ordering differs subtly from the contemporary definition:1 'For Cantor, M is well ordered by a relation if:
M is linearly ordered by <;
M has a <-first element, em>m0;
whenever N ⊆ M and ∃ m ∈ <>M - N ∀ n ∈ N [n < m] then there is a < - smallest m ∈ M - N such that ∀ n ∈ N [n < m] 2'
There are many different orderings for a set in elementary set theory. Such orderings can be expressed in terms of the relations or sets. One can say that a relation R on a set S is a certain kind of ordering, according to how it orders the elements of the set3. Alternatively, one can also describe a set as ordered in a certain way, according to the effect of the ordering relation R on the set. Referring to both, a simple ordering or chain is a set in which (if we focus, as is the custom, on binary relations and ordered pairs only) each element x has the relationship < with the next consecutive element y in the set. In other words, a set S is simple or chain ordered if x and y are elements of S and for every ordered pair (two elements selected from S between which the relation holds)  x,y in S, x<y (in other words, the relation in question is <, such that xRy.) Technically speaking, a simple ordering is anti-symmetric (xRy ≠ yRx), transitive (if xRy and yRz then xRz) and strongly connected (∀x∀y(x,y∈ S,xRy→yRx))4.
Example 1.
S = {1,2,3,4,5,6,7,8,9} is a simply ordered set
R = {1,2, 2,3, 3,4, 4,5, 5,6, 6,7, 7,8, 8,9} is a simple ordering relation applied to S
A partially ordered set is one which has a partial ordering. The relationship operator for numbers associated with a partial ordering of sets is  . Partial ordering is them same as simple ordering, except that for x and y being elements of S, x can be equal to y. Technically, a partial ordering R on S is reflexive (xRx), anti-symmetric (xRy≠yRx) and transitive (if xRy and yRz then zRz)5.
Example 2.
S = {1,1,2,5,5,5,7}
R = {1,1, 1,2,2,5,5,5,5,5,5,7}
A well-ordered set is, of course, defined by a well ordering (relation) on the set. A well ordering is simply a partial ordering in which there is a minimum or least element in every subset of S6. Thus, R in example 2. above is not a well ordering, and S is not a well-ordered set (due to the duplicate or repeated elements). The relation < applied to the set of natural numbers is a well ordering relation.
Example 3.
N = {1,2,3,4,5,6,7….n}
R = {1,2,2,3,3,4,4,5,5,6,6,7,7,8...x,y}
Note that each subset has a minimum, least or first element. Note also the similarity between the properties of the simple ordering or chain described above, and the properties of the well ordering. A well ordered set is in fact a special case of a simple ordered set, which is in turn a special case of a partially ordered set. That all sets, including sets with transfinite cardinalities, can be well ordered, is a pivotal assumption of Cantor's continuum hypothesis, which states that there is no set with cardinality falling between the cardinality of the set of finite countable ordinals and that of the countable infinite (transfinite) ordinals:
i.e. (I)  (II), and there can be no M such that (I) M (II) 7
where:
(I) is the cardinality (also called the power or enumeral) of the first number class, being the natural numbers8.
(II) is the cardinality of the second number class, being the infinite ordinal numbers9.
is the less power than or less 'pollent' than relational operator used for cardinalities10.
1. Michael Hallett, 'Cantorian Set Theory and Limitation of Size', Oxford: Oxford University Press, 1986, 51.
2. Ibid., 51.
3. Stoll, R. 'Set Theory and Logic', London and San Francisco: W.H. Freeman and Company, 1962, 48.
4. Suppes, P. 'Axiomatic Set Theory', Princeton, New Jersey: D. Van Nostrand Company Inc., 1960, 69, 72.
5. Ibid., 72.
6. Robert Stoll, op. cit., 53.
7. Michael Hallett, op. cit., 66.
8. Mary Tiles, 'The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise', London: Dover Publications, 1972, 104.
9. Ibid., 104.
10. Suppes., 258, 94.
Bibliography/References
Hallett, M. Cantorian Set Theory and Limitation of Size, Oxford: Oxford University Press, 1986, 51. Extract (75 words) from pp.51-52 from "Cantorian Set Theory and Limitation of Size" by Hallett, M. (1984). By permission of Oxford University Press.
Free permission
Stoll, R. Set Theory and Logic, London and San Francisco: W.H. Freeman and Company, 1962. (Dover Publications) By Permission.
Suppes, P. Axiomatic Set Theory, Princeton, New Jersey: D. Van Nostrand Company Inc., 1960. (Dover Publications) By Permission.
Tiles, M. The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise, London: Dover Publications, 1972. By Permission.
Book Resources On Well-ordered SetCantorian Set Theory and Limitation of Size by Hallett, M
The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise by Tiles, M
Set Theory and Logic by Stoll, R
Editor(s): Long, B.
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