MS 468-471 (1903) - Lowell Lecture V

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Then the exponential of the Ms will be the sam whose single members are these sams

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Now my proposition is that, in every case, the multitude of the exponential is greater than that of the primitive gath. That is, no matter what the Ms may be they are always fewer than all the possible sams of Ms. That is it is impossible to find any one to one relation such that every sam of Ms stands in that relation to an M. How am I going to prove this? I do it

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in this way: I say, imagine any one-to-one relation you please, and call this the relation r. Now I will exactly describe a sam of Ms which does not stand in that relation, r, to any M whatsoever. You might suppose it would be difficult to describe so exactly a sam of Ms for which this relation fails when we know nothing at all about this relation except that it is a one-to-one relation called r. Nevertheless I will describe a sam of Ms which does not stand in the relation r to any M at all. I will call it my test sam. This test sam shall be composed as follows. Whether or not it includes among its members any particular M, say M_x, depends upon what kind of a sam

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of Ms it is that is in the relation r to M_x. It may be that there is no sam of Ms in the relation r to M_x. In that case, I do not care whether M_x be included in my test sam or not. But if there be a sam of Ms that stands in the relation r to M_x then if this sam of Ms includes M_x, M_x shall be excluded from my test sam, while if the sam of Ms that stands in the relation r to M_x does not include M_x as a member of it, M_x shall be a member of my test sam. That describes precisely my test sam except as regards certain possible Ms which may be all included or all excluded or some included and some excluded as you please.

Anyway I say that this test-sam does not stand in the relation r to any M.

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For take any M you please, I care not what. Call this M_y. Now M_y belongs to one or other of three classes; namely 1st, the class of Ms which I positively require my test sam to include, or 2nd, the class of Ms which I require my test sam to exclude, or 3rd, the class of Ms as to which I do not care whether they be included in the test sam or not. If M_y belongs to the first class, it is one of the Ms that I require my test sam to contain. But those where the Ms that were not contained in the sams of M that were r to them. Plainly then if M_y belongs to this class my test sam is not r to it. For my test sam, in this case, contains M_y, while the only sam of Ms that is in the one-to-one relation r to M_y does not

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contain M_y. So this sole sam of Ms that is in the relation r to M_y is not my test sam. If M_y however belongs to the second class, it is one of those Ms which I expressly exclude from my test sam. It is however contained in the only sam of Ms that stands in the relation r to it. Plainly, then, this sam that alone is in the relation r to M_y if M_y belongs to the second class is not my test sam. The only remaining possibility is that M_y belongs to the third class. But in that case it is one of these Ms to which no sam of M stands in the relation r; so that my test sam cannot stand in that relation to it. Thus you perceive that to whatever class M_y belongs my test sam