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we may put ω((1),2). Then we shall have at length ω((1),(1),2) and so forth and then ω(((1))2) and so ω((((1)))2) ω(((((1))))2) and after that we may put w[2]. In short, there will be no end to the need of new symbols. It all follows from two principles, 1st, every number has another number next after it. 2nd, every endless series of numbers accurately describable in any manner whatever has a number next after it.
Cantor calls such a series a wohlgeordnet series. But I propose, in admiration of the genius that has discovered it, to call it a Cantorian succession.
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Cantor has conjectured that, given any collection whatever, any universe you please of independent quasi-individuals, there is a relation (and if one of course innumerable such) that in passing from relate to correlate and from correlate to correlates correlate, this relation arranges the whole universe in a Cantorian collection.
Indeed, Cantor put forward this as more than a conjecture,— as a consequence of an unacknowledged law of thought. But the proposition has been received by mathematicians with the gravest doubt.
If I have time, I will say more about this very important question later; but at present
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I will only say this. The relation is per se possible. It only supposes that certain individuals shall have special one-sided connections. Now every two individuals have special one sided connections. Neither can any limitation of existence render this form of relation self contradictory. Therefore, Cantor is right. This cannot be clear to you. It is the merest hint. But I might give you several lectures elucidating this matter. I must hurry on to other things.
Having made it clear what a collection is, the next thing I have to do is to define Multitude. If we regard plural nouns, such as men, horses, trolley-cars, as names describing sams
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men meaning a sam of which any member is a man, etc.: then the adjectives two, ten myriad, innumerable, express qualities of collections of a certain class, and any quality of this class is a multitude. Of course, this is not a definition. It is only the framework for a definition. In order to define multitude, it is necessary to begin by analyzing our meaning when we say that one collection is greater than another. This analysis was first published in a posthumous work of Bernard Bolzano, which appeared in 1851 and has since been reprinted Paradoxien des Unendlichen. Bolzano was a catholic theologian of Hungary and the author of a logic in four volumes. He was far
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too clearheaded to escape persecution in his position, you may be sure. Bolzano's definition amounts to this. If one gath, say that of the Bs, is so related to another gath, say the As, that there can be no one-to-one relation in which every B stands to an A, then, and only then, the gath of the Bs is greater than that of the As, and the latter is less than the former. [diagram with notes] For example there is no possible one-to-one relation in which every one of five things stands to one of three things.