by Corry Shores
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[Please see the original text, as I did not follow it closely.]
Bertrand Russell
Principles of Mathematics
Part 5: Infinity and Continuity
Ch.34: Limits and Irrational Numbers
Brief Summary:
In a compact continuum, there is always a number between any two others. A value to which a dense (‘compact’) continuum leads up to continuously is a limit.
Summary
§262
The concept of mathematical continuity depends on the concept of limit. It is no longer necessarily to appeal to the concept of the infinitesimal. [278]
We said that a compact series is “one in which there is a term between any two.” [278] But there are two classes of terms within such series that do not have terms between them. [To use our own example, if we have 1 and the successor function, then there is no term between them. This also makes 1 the limit of that series.]
But in such a series it is always possible to find two classes of terms which have no term between them, and it is always possible to reduce one of these classes to a single term. For example, if P be the generating relation and x any term of the series, then the class of terms having to x the relation P is one between which and x there is no term. The class of terms so defined is one of the two segments determined by x; the idea of a segment is one which demands only a series in general, not necessarily a numerical series. In this case, if the series be compact, x is said to be the limit of the class; when there is such a term as x, the segment is said to be terminated, and thus every terminated segment in a compact series has its defining term as a limit.
[278]
We will now define the limit. [Consider again the successor function. We have the numbers succeeding 1, and the numbers that 1 succeeds. What Russell is calling P would be the numbers coming before 1, or in his case x. x is the limit if there is always another number between it any number preceding it. ]
To obtain the general definition of a limit, consider any class u contained in the series generated by P. Then the class u will in general, with respect to any term x not belonging to it, be divisible into two classes, that whose terms have to x the relation P (which I shall call the class of terms preceding x), and that whose terms have to x the relation P˘ (which I shall call the class of terms following x). If x be itself a term of u, we consider all the terms of u other than x, and these are still divisible into the above two classes, which we may call πux and π˘ux respectively. If, now, πux be such that, if y be any term preceding x, there is a term of πux following y, i.e. between x and y, then x is a limit of πux. Similarly if π˘ux be such that, if z be any term after x, there is a term of π˘ux between x and z, then x is a limit of π˘ux. We now define that x is a limit of u if it is a limit of either πux or π˘ux. It is to be observed that u may have many limits, and that all the limits together form a new class contained in the series generated by P. This is the class (or rather this, by the help of certain further assumptions, becomes the class) which Cantor designates as the first derivative of the class u.
[279]
§263
Some traits of limits:
(1) they are normally found in compact series
(2) [the limit may be outside the series it is a limit of] “a limit may or may not belong to the class u of which it is a limit, but it always belongs to some series in which u is contained, and if it is a term of u, it is still a limit of the class consisting of all terms of u except itself.” [279]
(3) A class can have a limit only if it has an infinite number of terms.
(4) [Since each term is a limit to the series leading up to or leading after it:] “if u be co-extensive with the whole compact series generated by P, then every term of this series is a limit of u.” [279]
§264
Russell will now examine arithmetical theories of irrationals and their problems.
§265
We can divide rational numbers up such that an irrational lies between the classes. For example, square greater than and less than 2, with the square root of two being a number which does not fall within either class of rationals.
all rational numbers, without exception, may be classified according as their squares are greater or less than 2. All the terms of both classes may be arranged in a single series, in which there exists a definite section, before which comes one of the classes, and after which comes the other. Continuity seems to demand that some term should correspond to this section. A number which lies between the two classes must be a new number, since all the old numbers are classified. This new number, which is thus defined by its position in a series, is an irrational number. When these numbers are introduced, not only is there always a number between any two numbers, but there is a number between any two classes of which one comes wholly after the other, and the first has no minimum, while the second has no maximum. Thus we can extend to numbers the axiom by which Dedekind defines the continuity of the straight line (op. cit. p. 11):—
“If all the points of a line can be divided into two classes such that every point of one class is to the left of every point of the other class, then there exists one and only one point which brings about this division of all points into two classes, this section of the line into two parts.”
[281]
§266
Russell emends this formulation to:
A series, we may say, is continuous in Dedekind’s sense when, and only when, if all the terms of the series, without exception, be divided into two classes, such that the whole of the first class precedes the whole of the second, then, however the division be effected, either the first class has a last term, or the second class has a first term, but never both. This term, which comes at one end of one of the two classes, may then be used, in Dedekind’s manner, to define the section. In discrete series, such as that of finite integers, there is both a last term of the first class and a first term of the second class;* while in compact series such as the rationals, where there is not continuity, it sometimes happens (though not for every possible division) that the first class has no last term and the last class has no first term. Both these cases are excluded by the above axiom. But I cannot see any vestige of self-evidence in such an axiom, either as applied to numbers or as applied to space.
[282]
§267
Consider again the square root of 2. The rationals coming after it and those coming before it share as their limit the irrational between it, and they converge upon one another at that shared limit. [282-284]
§268
Weierstrass’ theory of irrationals is similar to Dedekind’s. [284]
Russell then notes objections to the above arithmetical accounts of irrationals:
Thus the arithmetical theory of irrationals, in either of the above forms, is liable to the following objections. (1) No proof is obtained from it of the existence of other than rational numbers, unless we accept some axiom of continuity different from that satisfied by rational numbers; and for such an axiom we have as yet seen no ground. (2) Granting the existence of irrationals, they are merely specified, not defined, by the series of rational numbers whose limits they are. Unless they are independently postulated, the series in question cannot be known to have a limit; and a knowledge of the irrational number which is a limit is presupposed in the proof that it is a limit. Thus, although without any appeal to Geometry, any given irrational number can | be specified by means of an infinite series of rational numbers, yet, from rational numbers alone, no proof can be obtained that there are irrational numbers at all, and their existence must be proved from a new and independent postulate.
Another objection to the above theory is that it supposes rationals and irrationals to form part of one and the same series generated by relations of greater and less. This raises the same kind of difficulties as we found to result, in Part II, from the notion that integers are greater or less than rationals, or that some rationals are integers. Rationals are essentially relations between integers, but irrationals are not such relations.
[284-285]
§269
Cantor notes that it is a logical error to try to deduce the limit from the series of which it is a limit. Cantor begins with the notion of fundamental series that are contained in larger series. They are either ascending or descending, and they are coherent if:
(1) both are ascending, and after any term of either there is always a term of the other;
(2) both are descending, and before any term of either there is always a term of the other; and if
(3) one is ascending, the other descending, and the one wholly precedes the other, and there is at most one term which is between the two fundamental series. [286]
The fundamental series of rationals is a denumerable series. In this series will be terms that we can subtract from one another to obtain their differences of value. Now, “given any number ε, however small, any two terms of the series which both come after a certain term have a difference which lies between + ε and − ε.” [286] Such a series can be of three kinds.
(1) [the series will be diminishing (to zero)] “any number ε being mentioned, the absolute values of the terms, from some term onwards, will all be less than ε, whatever ε may be” [286]
(2) [the series might be increasing] “from some term onwards, all the terms may be greater than a certain positive number ρ” [286]
(3) [the series will be diminishing through negative values] “from some term onwards, all the terms may be less than a certain negative number − ρ”. [286]
We now must define real number b. In the first case, b is zero. In the second case, be is positive, and in the third case b is negative. We now want to define operations such as addition for these numbers. So we must think of there being two such fundamental series, a and a’. In each series (which are in order), we may enumerate their order with subscripts. a1, a2, a3, a4, etc and a’1, a’2, a’3, a’4 . That would mean av, a’v, would be the vth number of both series. We now think of another series [which might be a part of the others?] whose vth term is av + a’v or av – a’v or av x a’v [so there would be a series of values that is the addition of the pairings of the other series, and subtraction, multiplication, etc.]. This series would also be a fundamental series. Now consider the series (av) + (a’v). In it will be real numbers b, b’. The numbers defined by (av + a’v) will be defined by b + b’, and similarly for the other operations [division included]. Now we need to define equal, greater, and less.
b = b' means b − b' = 0; b > b' means that b − b' is positive; and b < b' means that b − b' is negative
[287]
Now, we can even have a fundamental series with the same number repeating. If this were so, then a is the limit of this series. Russell thinks that we can say the following then:
connected with every rational a there is a real number, namely that defined by the fundamental series whose terms are all equal to a; if b be the real number defined by a fundamental series (aν) and if bν be the real number defined by a fundamental series whose terms are all equal to aν, then (bν) is a fundamental series of real numbers whose limit is b.
[287]
§270
“Hence there is no logical ground for distinguishing segments of rationals from real numbers.” [288, see text for more detail.]
Bertrand Russell. Principles of Mathematics. London/New York: Routledge, 2010 [1st published 1903].
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