The intuitive idea of a cardinal is to create some notion of the relative size or "bigness" of a set without reference to the kind of members which it has. For finite sets this is easy; one simply counts the number of elements a set has. In order to compare the sizes of larger sets, its necessary to appeal to more subtle notions.
A set Y is at least as big as a set X if there is a one-to-one mapping from the elements of X to the elements of Y. This is most easily understood by an example; suppose we have the sets X = {1,2,3} and Y = {5,6,7,8}, then using this notion of size we would observe that there is a mapping:
1 -> 5 2 -> 6 3 -> 7which is one-to-one, and hence conclude that Y has cardinality greater than or equal to X. The advantage of this notion is that it can be extended to infinite sets.
The classic example used is that of the infinite hotel paradox. Suppose you are a innkeeper at a hotel with an infinite number of rooms. The hotel is full, and then a new guest arrives. It's possible to fit the extra guest in by asking the guest who was in room 1 to move to room 2, the guest in room 2 to move to room 3, and so on, leaving room 1 vacant. In this way we can see that the set {1,2,3,...} has the same cardinality as the set {2,3,4,...} since a one-to-one mapping from the first to the second has been shown. This motivates the defintion of an infinite set being any set which has a proper subset of the same cardinality; in this case {2,3,4,...} is a proper subset of {1,2,3,...}.
It is provable that the cardinality of the real numbers is greater than that of the natural numbers just described. This is easily visualized using Cantor's diagonal argument; classic questions of cardinality (for instance the continuum hypothesis) are concerned with discovering whether there is some cardinal between some pair of other infinite cardinals. In more recent times mathematicians have been describing the properties of larger and larger cardinals.
Formally, the order among cardinal numbers is defined as follows: | X | <= | Y | means that there exists an injective function from X to Y. The Cantor-Bernstein-Schroeder theorem states that if | X | <= | Y | and | Y | <= | X | then | X | = | Y |. The axiom of choice is equivalent to the statement that given two sets X and Y, either | X | <= | Y | or | Y | <= | X |.
A set X is infinite?, or equivalently, its cardinal is infinite?, if there exists a proper subset Y of X with | X | = | Y |. It can then be proved that the finite cardinals are just the natural numbers, i.e., that a set X is finite if and only if | X | = | n | = n for some natural number n. It can also be proved that the cardinal ω of the set of natural numbers is the smallest infinite cardinal, i.e., that any infinite set admits a subset of cardinality ω. This first infinite cardinal is also often denoted by aleph-0 (where aleph is the first letter in the Hebrew alphabet). The next larger cardinal is denoted by aleph-1 and so on. For every ordinal a there is a cardinal number aleph-a.
If X and Y are disjoint, the cardinal of the union of X and Y is called | X | + | Y |. We also define the product of cardinals by | X | x | Y | = | X x Y | (the product on the right hand side is the cartesian product). Also | X || Y | = | XY | where XY is defined as the set of all functions from Y to X. It can be shown that for finite cardinals these operations coincide with the usual operations for natural numbers. For infinite sets (assuming the axiom of choice) we have | X | + | Y | = | X | x | Y | = max{| X |, | Y |}. On the other hand, 2| X | is the cardinality of the power set of the set X and Cantors Diagonal argument shows that 2| X | > | X | for any set X. This proves that there exists no largest cardinal. In fact, the class of cardinals is a proper class.
The continuum hypothesis states that there are no cardinals strictly between ω and 2ω (the cardinal of the set of real numbers, or the continuum, whence the name). The generalized continuum hypothesis states that there are no cardinals strictly between | X | and 2| X | for arbitrary infinite X. The continuum hypothesis is independent from the usual axioms of set theory, the Zermelo-Fraenkel axioms together with the axiom of choice (ZFC).
Most of the following axioms (in general, existence of large cardinals) are trivially satisfied by aleph-0, so always assume an implicit "larger then aleph-0" condition. There are deep reasons for this property of aleph-0.
An infinite cardinal k is strongly inaccessible if the following conditions hold:
It is provably impossible to prove the existence of strongly inaccessible cardinals in ZFC. It is an open question whether their non-existence can be proved; most people believe that it cannot. The same is true for all the classes of cardinals that follow:
A cardinal is 0-hyperinaccessible if it is an inaccesible cardinal.
A cardinal A is 1-hyperinaccessible if there are A inaccessibles less than A.
A cardinal B is (n+1)-hyperinaccessible if B is an n-hyperinaccesible preceeded by B n-hyperinaccessibles.
A cardinal C is hyper-hyperinaccessible (or hyper2-inaccessible) if C is n-hyperinaccessible for all n < C.
A cardinal D is 1-hyper2-inaccessible if there are D hyper2-inaccessible cardinals less than D.
A cardinal E is hyper3-inaccessible if E is n-hyper2-inaccessible for all n < E.
A cardinal F is super-hyperinaccessibles if F is preceeded by F hypern-inaccessibles for every n less than F.
A cardinal G which is preceeded by G inaccessible cardinals, G hyper-inaccessibles cardinals, G super-hyper-inaccessible cardinals, and so forth (continue the above construction) is called a Mahlo cardinal, after Mahlo who discovered them in 1911.
The existence of Mahlo cardinals cannot be proved to from the ZFC axioms by [Goedel Incompleteness Theorem]?, though they have not been proved not to exist either. They are useful in boolean relation theory.
Indescribable cardinals are cardinals much bigger than Mahlo cardinals.
Reference: http://logweb.terrashare.com/text/logic/infini.txt
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