Volume 1 / Part 5 — *216. DERIVATIVES.
Reading edition
*216. DERIVATIVES.
Summary of *216.
If is any class, and is any series, the derivative (or first derivative) of with respect to is the class of limits of existent sub-classes of , i.e. . That is, a term belongs to the derivative of if a set of terms exists which is contained both in and in , and has for its limit. The derivative of with respect to will be denoted by .
In general, there will be members of not contained in , and members of not contained in . is said to be dense in if all its terms except the first (if there is a first) belong to , that is, if all its terms except the first are limits of existent classes contained in . is said to be closed in if every existent sub-class of which has no maximum has a limit which belongs to , i.e. if every existent sub-class of has a limit or a maximum, and the derivative of is contained in . If is both dense and closed, it is called perfect. In this case, all its terms are limits of classes chosen out of , and every class chosen out of has a limit or maximum in .
The second derivative of is , i.e. , and so on. (Derivatives of infinite order cannot be dealt with till a later stage.) If is serial, the second derivative of is always contained in the first (*216·14).
If is a Dedekindian series, is closed whenever . In order to secure a Dedekindian series, it is sometimes convenient to replace by the ordinally similar series , which is contained in the Dedekindian series . Then is replaced by , and is closed if the derivative of with respect to is contained in . The relation of the derivative of in to the derivative of in has been treated in *212·6 and following propositions. This subject is resumed below (*216·5 ff.).
The derivative of the series P will be defined as the series of its limit-points, and denoted by . Thus we put
If is a series, the derivative of a class consists of those members of which are such that members of exist in every interval which ends in , i.e.
*216·13.
We have
*216·2.
*216·3.
*216·32.
We prove (*216·4—·412) that the properties of with respect to , as regards being dense, closed, or perfect, belong to with respect to if is a correlator of with .
We next consider the relation of in to in (*216·5—·56). The point of these propositions is that is Dedekindian, so that a class is closed in if it contains its first derivative. (It is usual to define a class as closed whenever it contains its first derivative; but this involves the tacit assumption that the series is Dedekindian. If is the series of real numbers, this assumption is of course verified.) We prove (*216·52) that the derivative of in is , i.e. is the class of segments defined by such existent sub-classes of as have no maximum; we show that is dense, closed, or perfect in according as is dense, closed, or perfect in (*216·53·54·56), and that and are closed if contains its first derivative (*216·54).
We end with various propositions on (*216·6—·621), of which the chief is
*216·611.
This subject will be resumed in connection with well-ordered series in *264.
*216·01.
*216·02.
*216·03.
*216·04.
*216·05.
*216·1.
*216·101.
Dem.
*216·11.
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*216·111.
*216·12.
*216·13.
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*216·14.
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*216·15.
*216·16.
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*216·2.
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*216·21.
*216·22.
*216·23.
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*216·3.
*216·31.
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*216·32.
*216·33.
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*216·34.
*216·35.
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*216·36.
*216·37.
*216·371.
*216·38.
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*216·381.
*216·382.
*216·4.
Dem.
*216·401.
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*216·41.
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*216·411.
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*216·412.
*216·5.
Dem.
*216·51.
Dem.
*216·52.
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*216·521.
Dem.
*216·53.
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*216·54.
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*216·55.
Dem.
*216·56.
*216·6.
*216·601.
Dem.
*216·602.
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*216·603.
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*216·61.
*216·611.
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*216·612.
Dem.
*216·62.
Dem.
*216·621.