Lecture
TRANSFINITE NUMBER (from trans... and Latin finitus – bounded), a generalization of the notion of an ordinal number (see below). The definition of a transfinite number relies on the notion of a well-ordered set. Every finite set can be made well-ordered by arranging all its elements in a certain order. The simplest example of an infinite well-ordered set is the set of all natural numbers arranged in increasing order; the same set arranged in decreasing order (so that the larger is considered to precede the smaller) will no longer be well-ordered, since none of its infinite subsets has a first element. Two ordered subsets X and Y are called similar, or having the same order type, if a one-to-one correspondence preserving the order of elements can be established between their elements. All finite well-ordered sets containing the same number of elements are similar to one another. Therefore the order types of finite well-ordered sets can be identified with the natural numbers, which thus appear as ordinal numbers (whereas, characterizing the number of elements of a set, the same natural numbers appear in their other aspect – as cardinal numbers).
A transfinite number is what the order types of infinite well-ordered sets are called. Thus the notion of a transfinite number is an extension of the notion of an ordinal number to infinite sets. An analogous generalization of the cardinal number leads to the notion of the cardinality of a set. Since sets of unequal cardinality cannot be put into one-to-one correspondence, well-ordered sets of different cardinality correspond to different transfinite numbers. However the converse (unlike for finite sets) is false: infinite well-ordered sets may be equinumerous without being similar, and thereby define different transfinite numbers.
For transfinite numbers one can introduce the notions of «greater» and «smaller». Namely, a transfinite number α, by definition, is smaller than a transfinite number β (α<β) if some (and hence any) well-ordered set of type α is similar to some initial segment of some (and consequently any) set of type β. Here an initial segment of a well-ordered set, cut off by an element x, is the subset of its elements that precede x. For any two transfinite numbers α and β always either α<β, or α=β, or α>β.
In the application of transfinite numbers to various questions of mathematics, an important role is played by the principle of transfinite induction, which generalizes the ordinary principle of mathematical induction to arbitrary well-ordered sets: if some proposition is true for the first element of a well-ordered set X and if from the fact that it is true for all elements of the set X preceding the given element x of the set X, its validity follows also for the element x, then this proposition is true for every element of the set X.
In mathematics , transfinite numbers are numbers that are « infinite » in the sense that they are greater than all finite numbers, but not necessarily absolutely infinite . They include the transfinite cardinals , which are cardinal numbers, used to quantify the size of infinite sets, and transfinite ordinals , which are ordinal numbers, used to order infinite sets. The term transfinite was introduced by Georg Cantor in 1915, who wished to avoid some of the implications of the word infinite in connection with these objects which, nevertheless, were not finite . Few modern writers share these doubts; it is now accepted to call transfinite cardinals and ordinal numbers "infinite". Nevertheless, the term «transfinite» also remains in use.
Any finite number can be used in at least two ways: as an ordinal and as a cardinal. Cardinal numbers specify the size of sets (for example, a bag of five marbles), whereas ordinal numbers specify the order of members in an ordered set (for example, «third from the left» or «the twenty-seventh day of January."). When extended to transfinite numbers, these two notions become different. A transfinite cardinal number is used to describe the size of an infinitely large set, and a transfinite ordinal number is used to describe a location within an infinitely large ordered set. The most notable ordinal and cardinal numbers are, respectively:
( Omega ): the smallest transfinite ordinal number. It is also the order type of the natural numbers in their usual linear order.
( Aleph-null ): the first transfinite cardinal number. It is also the cardinality of the infinite set of natural numbers. If the axiom of choice holds, the next higher cardinal number is aleph-one; if not, there may be other cardinals that are incomparable with aleph-one and greater than aleph-null. In any case, there are no cardinals between aleph-null and aleph-one.
The continuum hypothesis is the conjecture that there are no cardinal numbers intermediate between aleph-null and the cardinality of the continuum (the cardinality of the set of real numbers): or, equivalently, that the cardinality of the set of real numbers is aleph-one. In Zermelo–Fraenkel set theory neither the continuum hypothesis nor its negation can be proved without violating consistency. 
Some authors, including P. Suppes and J. Rubin, use the term transfinite cardinal to denote the cardinality of a Dedekind-infinite set in contexts where this may not be equivalent to «infinite cardinal»; that is, in contexts where the axiom of countable choice is not assumed or is not known to hold. Given this definition, all of the following are equivalent:
is a transfinite cardinal. That is, there exists a Dedekind-infinite set whose cardinality is
In Cantor's theory of ordinal numbers, every integer must have a successor. The next integer after all the ordinary ones is named, that is, the first infinite integer. In this context there are larger ones, and larger still. Arithmetic expressions containing an ordinal designation can be regarded as the set of all integers up to that number. A given number usually has several expressions that represent it, but there is a unique Cantor normal form that represents it, essentially a finite sequence of digits that gives the coefficients of decreasing powers. 
However, not all infinite integers can be represented by the Cantor normal form, and the first form that cannot be represented is called the limit. It is the smallest solution for it, and the following solutions give even larger ordinal numbers, and they can be tracked until a limit value is reached, which is the first solution for it. This means that in order to be able to define all transfinite integers, one must devise an infinite sequence of names: because, if one were to name a single largest integer, then one could always mention its larger successor. But, as Cantor observed, even this reaches only the lowest class of transfinite numbers: those whose set sizes correspond to the cardinal number. 
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