*Class* may be defined either extensionally or intensionally. That is to say, we may define the kind of object which is a class, or the kind of concept which denotes a class: tihs is the precise meaning of the opposition of extension and intension in this connection. But although the general notion can be defined in this two-fold manner, particular classes, except when they happen to be finite, can only be defined intensionally, i.e. as the objects denoted by such and such concepts. I believe this distinction to be purely psychological: logically, the extensional definition appears to be equally applicable to infinite classes, but practically, if we were to attempt it, Death would cut short our laudable endeavour before it had attained its goal. Logically, therefore, extension and intension seem to be on a par. I will begin with the extensional view.(§ 71 ¶ 1)

When a class is regarded as defined by the enumeration of its terms, it is more naturally called a *collection*. I shall for the moment adopt this name, as it will not prejudge the question whether the objects denoted by it are truly classes or not. By a collection I mean what is conveyed by

or `A` and `B`

or any other enumeration of definite terms. The collection is defined by the actual mention of the terms, and the terms are connected by `A` and `B` and `C`,*and*. It would seem that *and* represents a fundamental way of combining terms, and that just this way of combination is essential if anything is to result of wihch a number other than 1 can be asserted. Collections do not presuppose numbers, since they result simply from the terms together with *and*: they could only presuppose numbers in the particular case where the terms of the collection themselves presuppose numbers. There is a grammatical difficulty which, since no method exists of avoiding it, must be pointed out and allowed for. A collection, grammatically, is singular, whereas `A` and `B`, `A` and `B` and `C`, etc. are essentially plural. This grammatical difficulty arises from the logical fact (to be discussed presently) that whatever is many in general forms a whole which is one; it is, therefore, not removable by a better choice of technical terms.(§ 71 ¶ 2)

The notion of *and* was brought into prominence by Bolzano^{[57]}. In order to understand what infinity is, he says, we must go back to one of the simplest conceptions of the understanding, in order to reach an agreement concerning the word that we are to use to denote it. This is the conception which underlies the conjunction

(§ 71 ¶ 3)*and*, which, however, if it is to stand out as clearly as is required, in many cases, both by the purposes of mathematics and by those of philosophy, I believe to be best expressed by the words A system (Inbegriff) of certain things,

or a whole consisting of certain parts.

But we must add that every arbitrary object `A` can be combined in a system with any others `B`, `C`, `D`, ..., or (speaking still more correctly) already forms a system by itself^{[58]}, of which some more or less important truth can be enunciated, provided only that each of the presentations `A`, `B`, `C`, `D`, … in fact represents a *different* object, or in so far as none of the propositions `A` is the same as `B`,`A` is the same as `C`,

etc., is true. For if e.g. `A` is the same as `D`,`A` is the same as `B`, then it is certainly unreasonable to speak of a system of the things `A` and `B`.

The above passage, good as it is, neglects several distinctions which we have found here necessary. First and foremost, it does not distinguish the many from the whole which they form. Secondly, it does not appear to observe that this method of enumeration is not practically applicable to infinite systems. Thirdly, and this is connected with the second point, it does not make any mention of intensional definition nor of the notion of a class. What we have to consider is the difference, if any, of a class from a collection on the one hand, and from the whole formed of the collection on the other. But let us first examine further the notion of *and*.(§ 71 ¶ 4)

Anything of which a finite number other than 0 or 1 can be asserted would be commonly said to be many, and many, it might be said, are always of the form

Here `A` and `B` and `C` and ....`A`, `B`, `C`, ... are each one and are all different. To say that `A` is one seems to amount to much the same as to say that `A` is not of the form

To say that `A _{1}` and

is meaningless, so that diversity is implied byAandA

A term `A` which is one may be regarded as a particular case of a collection, namely as a collection of one term. Thus every collection which is many presupposes many collections which are each one: * A and B* presupposes

is one, but presupposesAdiffers fromB

Every pair of terms, without exception, can be combined in the manner indicated by * A and B*, and if neither

The question may now be asked: What is meant by * A and B*? Does this mean anything more than the juxtaposition of

we can replace the proposition byAandBare yellow,

andAis yellow

; but this cannot be done forBis yellow

; on the contrary,AandBare two

As regards what is meant by the combination indicated by *and*, it is indistinguishable from what we before called a numerical conjunction. That is, * A and B* is what is denoted by the concept of a class of which

Ais au,

are true, but of which all other propositions of the same form are false, thenBis au

allis the concept of a class whose only terms areu's

AandB

are the class, but are distinct from the class-concept and from the concept of the class.(§ 71 ¶ 9)AandB

The notion of *and*, however, does not enter into the *meaning* of a class, for a single term is a class, although it is not a numerical conjunction. If `u` be a class-concept, and only noe proposition of the form

be true, then `x` is a `u`all

is a concept denoting a single term, and this term is the class of which `u`'sall

is a concept. Thus what seems essential to a class is not the notion of `u`'s*and*, but the being denoted by some concept of a class. This brings us to the intensional view of classes.(§ 71 ¶ 10)

§ 71 n. 1. Paradoxien des Unendlichen, Leipzig, 1854 (2nd ed., Berlin, 1889), § 3. ↩

§ 71 n. 2. i.e. the combination of `A` with `B`, `C`, `D`, ... already forms a system. ↩

The Principles of Mathematics was written by Bertrand Russell, and published in in 1903. It is now available in the Public Domain.