Transfinite Numbers
Transfinite numbers are infinite ordinal numbers. Informally, ordinal numbers may be represented by strings of *'s. For example, the ordinal number 0 is the string of no stars. Number 1 is *. Number 2 is ** and so on. The first transfinite number is then represented by ***.... Here the three dots after the *** imply that the sequence repeats infinitely. This number is traditionally called omega and is written . Ordinal numbers are added by juxtaposition. For example 1 + 2 = (*) + (**) = *** = 3. So 1 + = ****... = . But + 1 = ****...*. These two numbers are different for the following reason. If a frog is on the first star of ***..., and it can leap onto successive stars, then it can reach any star in ***.... But, a frog on the first star of ***...* cannot get to the last star by hopping consecutive stars. So, addition of transfinite numbers is not commutative.
Suppose that x and y are two transfinite numbers. Then x times y is represented by replacing every * in y by a copy of x. For example, 2 times is ******... = ***... = . But times 2 is ***...***.... So multiplication of transfinite numbers is not commutative either. Exponentiation is harder to define so consider these examples. In order to make the notation easy, we will write x{y}instead of xy.
- 2{} = [**]... =
- {2} = [***...]...
- {3} = [[***...]...]...
- ( +1){2} = [***...*]...[***...*] = [***...]...[***...*] = {2} + 1.
The three dots after a pair of square brackets means that everything in the square brackets is repeated an infinite number of times. To get {}, the pattern started by {2} and {3} must be repeated times. Notice that times {} = {}. The number {{{...}}}}, that is raised to the power of raised to the power of and so on times, is commonly denoted by e0. It is the first ordinal number that cannot be obtained by a finite number of additions, multiplications, and exponentiations. For this reason, it is called the first inaccessible number.
Formally, ordinal numbers are defined in terms of well-ordered sets and order types. A set W is well ordered if there is a relation (commonly denoted by ) on it that satisfies these five properties:
- (1) for every w in W, w w
- (2) if w v and v x then w x
- (3) for every pair of elements w and v in W, either w v or v w
- (4) there is an element (called the least element) l such that if w is any element of W then l w.
- (5) if w is in W then there is an element v (called the successor of w) such that w v and if x is any element not equal to w and w x then v ; x.
For example, the positive integers are well-ordered but the positive rational numbers are not since no element has a successor. The words of the English language are well-ordered by "dictionary" order. In this case, the least element is 'a'. If V and W are well-ordered sets and there is a one-to-one correspondence f from V to W such that v ; w if and only if f(v) ; f(w) then V and W are said to be of the same order type. An ordinal number is an order type of a well-ordered set.
If x and y are two ordinal numbers then they are represented by two well-ordered sets X and Y whose order type is x and y respectively. Then x + y is defined to be the order type of the set Q = X Y with the following order relation. If v and w are both in X and v ; w as elements in X, then v ; w as elements of Q. A similar statement holds for Y. If v is an element of X and w is an element of Y, then x ; w.
x times y is defined to be the order type of the set P, of all ordered pairs of the form (v, w) in which v is an element of X and w is an element of W with the following order relation. If (v, w) and (a, b) are elements of P then (v, w) ; (a, b) if either v < a or v = a and w ; b. This is sometimes called "dictionary" order.
x to the power of y is defined to be the order type of the set R of all functions from X to Y with the following order relation. If f and g are in R, then f ; g means that if v is the first element v of X such that f(v) is not equal to g(v), then f(v) ; g(v). For example, 2{} is represented by the set of all functions the set {0,1} to the positive integers.
If x and y are ordinal numbers then x y means that there is function f from X to Y that maps the least element of X to the least element of Y and if w is the successor of v then f(w) is the successor of f(v). Therefore x is represented by the subset f(X) of Y. For example 0 < 1 < 2 < ... < < + 1 < ... < times 2 < ... < {} < e0 <....
So the ordinal numbers can all be represented by sets having the property that x y means X Y. Thus the union of all these sets has an order type (called P) that is larger than any ordinal number. But then P + 1 is an ordinal larger than P. This contradiction is called the Burali-Forti paradox. Many mathematicians were suspicious of transfinite numbers for this reason. Set theorists resolved this paradox by stating that the union of all those sets is not a set itself. It is a class of objects, and thus, not an ordinal number.
Cantor's student Zermelo proved that every set can be well-ordered if the Axiom of choice is true. In other words, the ordinal numbers can be used to "count" the elements of any set. The axiom of choice was later proved to be equivalent to Zorn's lemma.
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