MS 468-471 (1903) - Lowell Lecture V

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false. Adopting it for the moment, we see that the smallness of a collection compels descriptions of identities that in a larger collection are not compelled. But this is as much as to say that propositions follow as necessary consequences from premisses relating to members of smaller collections when in a larger collection no such conclusion would follow. If one collection is larger than another this will invariably be true. From this point of view you see the immense logical significance of multitude.

For instance, suppose you want to find out who committed certain murders, who some Jack the Ripper is. Plainly if you can ascertain will be sure that he belongs to a certain small collection of persons

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your investigation may be greatly helped. You have ways of reasoning open to you that you had not before. Indeed if you can reduce the multitude of that collection to unity, your problem is solved.

Let us see then what all the different grades of multitude are. The lowest is unity for Nullity is not a multitude. Unity is that quality of a collection which consists in the absence of all diversity of its members. Add a single diverse individual to a collection of one, and you get a larger collection, and this is true, not only of a collection of one but ...

... counting is performed by successive movements

Let me prove this by graphs. In each graph which I write, the universe of discourse with be the members of the gath I am dealing it. Then nullity will be expressed by [] which is of course the pseudograph. For a universe with nothing in it is absurd. I add one to this. That is I express that there is something such that if anything is different from this the graph of nullity holds for it. That is [] or []. It is the graph of Unity. I add a unit to this. That is I express that there is something other than what the graph of unity asserts, but for all that is not this that graph holds. This gives []. It is the graph of Twoness. I again add a unit, asserting that there is something else but that apart from this new unit the graph of Twoness holds. This gives [] which is the graph of Threeness. In order to express that the gath taken as the universe is greater than a given number you must assert all the diversity that the graph of that number asserts and also the diversity it denies. That is, we must remove the large cut. Then in order to assert that our gath is not

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We call a collection of the members of which the Syllogism of Transposed Quantity necessarily holds an enumerable collection and one of which this is not true or where there is a Fermatian relation a denumeral collection,— in German abzählbar, of one, it follows that it is true of every enumerable collection.

All the whole numbers, properly so called, that is, all numbers of which our system of so called Arabic notation affords a definite symbol,— all the numbers up to any one form an enumerable collection. But the entire collection of whole numbers capable of representation in that system is a collection not enumerable, but innumerable. And the single multitude of all the whole numbers, or of any such endless series of which all the members up to any member form an enumerable collection is called the denumeral collection, “abzählbar” in German. The denumeral multitude is a single multitude of the class of innumerable multitudes.

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more than the number we must enclose the last graph in a cut. Thus to express that that gath in question is more than three we have the graph []. To assert that it is not more than three we scribe []. To assert that it is neither three nor more than three, that is that it is less than three, we scribe [diagram]. We now iterate the right hand graph in the second cut of the left hand one thus [diagram]. We extend the ligature by the rule of Iteration and join them inside 3 cuts by the Rule of Insertion. We thus get, after erasing the left hand part