Chains. If the elements of A are ele ments of B, A is part of B. If in this case not all b's are a's, A is a proper part of B. Any assemblage is part, but not proper part, of itself. One of the most important ideas con nected with that of the depiction of an assem blage on itself is the notion of a chain: if A be depicted on itself in any definite way, then any part of A that consists of the pictures thus formed is a chain. The theory of chains, due to Dedekind, has fundamental bearings in logic.
See INDUCTION, MATHEMATICAL.
The Concept of Equivalence or Sameness of Power. If A and B are such that each may be similarly depicted on the other, i.e., if a one-to-one correspondence can be established between the elements of A and those of B, A and B are said to be equivalent or to have the same power (.111iichtigkeit), a relation sym bolically expressed by writing A B or B A. Thus if A denote the assemblage of positive integers and B denote, say, the even positive integers, A B, for plainly one may pair 1 with 2, 2 with 4, 3 with 6 and so on. Other ways of pairing A and 13 in this case will readily occur to the reader.
Distinction of Finite and Infinite.An assemblage is infinite if it has the same power as some proper part of itself. Thus A of the last example is infinite. So, too, is B, for it is easily seen that if A B and if either A or B is infinite, so is the other. The foregoing defi nition of infinite is one of the most fruitful of modern concepts. It is due independently to Dedekind and Georg Cantor. A finite assem blage is one which can he exhausted by the re moval of its elements one at a time. Some times an infinite assemblage is defined to be one that cannot be exhausted or emptied by remov ing from it one element at a time. It has not been proved that the two definitions are logi cally equivalent. All apparent proofs of this depend on the questionable axiom of Zermelo (see below). For the purposes of investiga tion, the former is found to be by far the bet ter instrument. An infinite assemblage is often described as trans finite.
Denurnerability. Let A denote the assem blage of positive integers. Any assemblage B such that B A is said to be denumerable or to have the power of the denumerable assem blage. As A ^ A, A is itself denumerable, and it serves conveniently as the type of denumer able assemblages. The domain of such assem blages is exceedingly rich and is replete with surprises. For example, though the rational fractions, that is, fractions having integral terms, are so numerous that between any two of them, however near to each other in value, there is an infinity of others, nevertheless the assemblage of rational fractions including the integers is denumerable. Of this the reader can quickly convince himself by reflecting that there is but a finite number of such fractions of which each has a specified integer n for sum of its terms. Thus, if n=2, one has 1 or I ; if n = 3, one has i and t; if is = 4, I, f; and so on. Some are repeated ; repetitions may be kept or rejected. Dropping them, the required equivalence is seen in the pairing: (1, 1.) (2, I), (3,0; (4,0, (5,f), (6,3); In ordinary speech one is justified in saying that rational numbers are neither more nor less numerous than the integers or than the odd or the even integers. It is plain that the familiar
axiom, the whole is greater than any of its parts, is not valid for infinite assemblages. For finite assemblages it is valid absolutely ; whether for others is not known. For another example, consider the algebraic numbers before men tioned. These include the rationals and in finitely many besides. Nevertheless the assem blage of all algebraic numbers is denumerable. The proof is too long to insert here. Yet more astonishing is the theorem that an assemblage composed of all the elements of a denumerable infinity of denumerably infinite assemblages is denumerable.
The Power of the Continuum. At this stage the query is natural: is every possible as semblage denumerable? The answer is, no. The assemblage of all real numbers, i.e., of all rationals and irrationals, is said to constitute a continuum. The assemblage of points of a straight line is a continuum, in particular a linear continuum. The last two assemblages are in fact of the same power, but neither is denumerable. This is demonstrated by letting ca, ., . represent any denumerable as semblage of real numbers and then proving that between any two arbitrarily assumed numbers a and p there is one number and therefore an infinity of numbers not in the given sequence. From this proposition of Cantor s the existence of transcendental numbers, which had been otherwise previously proved by Lionville, fol lows as a corollary. Any assemblage equiva lent to that of the real numbers or to that of the points of a straight line is said to have the power of the continuum. The assemblage of points of any line-segment however short or, i what is the same, the assemblage of numbers between any two numbers however nearly equal, has the power of the continuum. In deed, either of these assemblages is a con tinuum. But an assemblage may have the power of the continuum without being a continuum. For example, the assemblage of transcendental numbers, though it is not a continuum, has the power of a continuum. In fact, the assemblage left on suppressing from a continuum any de numerable assemblage of elements is equivalent to the original assemblage. This last is a special case of the proposition: If A be infinite, and if the remainder R on suppressing a de numerable part of A be infinite, then R A. As above seen the power of the continuum is higher than that of the denumerable assemblage, but whether it is the next higher is an outstand ing question. There are higher powers than that of any given power (for there is no assem blage of all things), but no assemblage of points has a power higher than that of a continuum. On the contrary, it is one of the most marvelous of known facts that the assemblage of points on a line-segment however short is equivalent to the assemblage of all the points of space, nay, is equivalent to all the points of a space having not merely, like our own, three dimensions, but a denumerable infinity of dimensions.