The analogy of the contradiction of Burali-Forti with the contradiction involved in the notion of an “unknowable” may be set forth as follows. If A should say to B: “I know things which you never by any possibility can know,” he may be speaking the truth. In the same way, ω may be said, without contradiction, to transcend all the finite integers. But if some one else, C, should say: “There are some things which no human being can ever know anything about,” he is talking nonsense.[91] And in the same way if we succeeded in imagining a number which transcends all numbers, we have succeeded in imagining the absurdity of a number which transcends itself.
All the paradoxes of logic (or “the theory of aggregates”) are analogous to the difficulty arising from a man’s statement: “I am lying.”[92] In fact, if this is true, it is false, and vice versa. If such a statement is spread out a little, it becomes an amusing hoax or an epigram. Thus, one may present to a friend a card bearing on both sides the words: “The statement on the other side of this card is false”; while the first of the epigrams derived from this principle seems to have been written by a Greek satirist:[93]
Lerians are bad; not some bad and some not;
But all; there’s not a Lerian in the lot,
Save Procles, that you could a good man call;—
And Procles—is a Lerian after all.
This is the original of a well-known epigram by Porson, who remarked that all Germans are ignorant of Greek metres,
All, save only Hermann;—
And Hermann’s a German.
[88] Md., N. S., vol. iv., 1895, p. 168.
[89] First Principles, 6th ed., 1900, pp. 107-10. The first edition was published in 1862.
[90] Note on p. 6 of his paper: “On Infinity; and on the Sign of Equality,” Trans. Camb. Phil. Soc., vol. xi., part i., pp. 1-45 (read May 16, 1864).
[91] The assertion of the finitude of a man’s mind appears to be nonsense; both because, if we say that the mind of man is limited we tacitly postulate an “unknowable,” and because, even if the human mind were finite, there is no more reason against its conceiving the infinite than there is for a mind to be blue in order to conceive a pair of blue eyes (cf. De Morgan, loc. cit.).