Volume 1 / Part 5 — *207. LIMITS.
Reading edition
*207. LIMITS.
Summary of *207.
A term is said to be the "upper limit" of in if has no maximum and is the sequent of . In this case, immediately follows the class , though there is no one member of which immediately follows. Sequents which are limits have special importance, and it is convenient to have a special notation for them. We write "" for the upper limit of ; or, if it is more convenient, "". (This is more convenient when is replaced by an expression consisting of several letters, or by a letter with a suffix.) The lower limit of will be the immediate predecessor of when has no minimum; this we denote by .
The following propositions on limits for the most part follow immediately from the propositions of *206 on sequents.
Our definition is so framed that the limit of the null-class is the first member of our series (if any). This departure from usage is convenient in order that, whenever our series contains any limiting point in the ordinary sense, the series of limiting points may exist, i.e. in order that may exist whenever there are existent parts of CʻP which have upper limits. The series is the "first derivative" of . The definition of a limit is
Besides the limit, we require, for many purposes, a single notation for the "limit or maximum." This we denote by "," putting Similarly for the lower limit or minimum we use "," putting We have (*207·101) and (*207·401). Hence it is unnecessary to prove propositions concerning lower limits, since they result immediately from propositions concerning upper limits.
In virtue of our definition of a limit, limits if is a sequent of and has no maximum (*207·1). Thus if has a maximum, it has no limit (*207·11), but if it has no maximum, the class of its limits is the class of its sequents (*207·12). Thus the existence of the class of limits is equivalent to the existence of the class of sequents combined with the non-existence of the class of maxima, i.e.
*207·13.
*207·2—·232 consist of various formulae for . We have
*207·2.
I.e. the whole of precedes , but any predecessor of precedes some member of .
*207·231.
I.e. the limit of , if it exists, is the term whose predecessors are identical with the predecessors of some part of .
We have also
*207·232.
This proposition should be compared with *205·54, which (slightly re-written) is
From the two together we arrive at
*207·51.
which serves to illustrate the utility of "."
We have
*207·24.
I.e. if is connected, a class cannot have more than one limit; also
*207·25.
I.e. any terms which have some 's beyond them may be added to without altering the limit.
We next have a set of propositions (*207·251—·27) proving that if a class has a limit, any single term of the class may be removed without altering the limit (*207·261), and that in any case, provided the class is not a unit class, its minimum (if any) may be removed without altering the limit (*207·27). We then prove (*207·291) that if is a series, and is a class which has a limit, the predecessors of the limit are the class .
We then have a set of propositions (*207·3—·36) on the limit of and kindred matters. If has no immediate predecessor, the limit of is , and vice versa (*207·32·33). Hence
*207·35.
I.e. the limit-points of are those which have no immediate predecessors.
We next turn our attention to "." This again is one-many, provided is connected (*207·41). We have by the definition
*207·42.
*207·43.
*207·44.
*207·45.
Also we have
*207·46.
which is a very useful proposition, as is also *207·51 (given above).
A useful proposition in dealing with classes of classes contained in a series is
*207·54.
I.e. if every member of has a limit, the limit or maximum (if any) of the limits is the limit or maximum, and in fact the limit, of .
We have next a set of propositions (*207·6—·66) on correlations, proving that the limit, or the , of the correlates is the correlate of the limit or , i.e.
*207·6.
*207·64.
The last three propositions (*207·7—·72) are lemmas for use in the theory of stretches (*215·5·51).
*207·01.
*207·02.
*207·03.
*207·04.
*207·1.
*207·101.
We shall not give further propositions on lower limits, unless for some special reason, since all of them result from propositions on upper limits by means of *207·101.
*207·11.
*207·12.
*207·121.
*207·13.
*207·14.
The above proposition is important because is the characteristic of "Dedekindian" series, i.e. of such as fulfil Dedekind's axiom.
*207·15.
*207·16.
*207·17.
*207·18.
Dem.
*207·2.
*207·21.
Dem.
*207·22.
This is very often the most convenient form for . It states that a limit of is a member of such that wholly precedes , but every predecessor of precedes some member of .
*207·23.
Dem.
*207·231.
*207·232.
*207·24.
Dem.
*207·25.
Dem.
*207·251.
Dem.
*207·26.
*207·261.
*207·262.
*207·263.
*207·27.
Dem.
*207·28.
*207·281.
*207·282.
*207·29.
Dem.
*207·291.
Dem.
*207·3.
Dem.
*207·31.
Dem.
*207·32.
*207·33.
*207·34.
Dem.
*207·35.
Dem.
*207·36.
Dem.
In virtue of this proposition, all limits are limits of classes of the form . In this respect, limits (in general) differ from segments. If we call the segment defined by , there will in general be segments not of the form . These, however, will be the segments which have no sequents, and therefore no limits; thus their existence does not introduce limits not derivable from classes of the form .
*207·4.
*207·401.
*207·41.
*207·42.
*207·43.
*207·44.
*207·45.
*207·46.
Dem.
*207·47.
Dem.
*207·48.
*207·481.
*207·482.
Dem.
*207·5.
*207·51.
*207·52.
*207·521.
Dem.
*207·53.
Dem.
*207·54.
Dem.
*207·55.
*207·6.
Dem.
*207·61.
*207·62.
*207·63.
Dem.
*207·64.
*207·65.
*207·66.
*207·7.
Dem.
*207·71.
*207·72.