> For example, Wittgenstein understood perfectly > well how to apply Cantor's argument and he > certainly is not thought of as believing in > a completed infinity. > > However, he also did not attack the mathematicians > who conducted investigations along those lines.
There is no path to infinity, not even an endless one. [§ 123]
It isn't just impossible "for us men" to run through the natural numbers one by one; it's impossible, it means nothing. [?] you can?t talk about all numbers, because there's no such thing as all numbers. [§ 124]
There's no such thing as "all numbers" simply because there are infinitely many. [§ 126]
Generality in mathematics is a direction, an arrow pointing along the series generated by an operation. And you can even say that the arrow points to infinity; but does that mean that there is something ? infinity ? at which it points, as at a thing? Construed in that way, it must of course lead to endless nonsense. [§ 142]
If I were to say "If we were acquainted with an infinite extension, then it would be all right to talk of an actual infinite", that would really be like saying, "If there were a sense of abracadabra then it would be all right to talk about abracadabraic sense perception". [§ 144]
Set theory is wrong because it apparently presupposes a symbolism which doesn't exist instead of one that does exist (is alone possible). It builds on a fictitious symbolism, therefore on nonsense. [§ 174]
Imagine set theory's having been invented by a satirist as a kind of parody on mathematics. ? Later a reasonable meaning was seen in it and it was incorporated into mathematics. (For if one person can see it as a paradise of mathematicians, why should not another see it as a joke?)
If it were said: "Consideration of the diagonal procedure shews you that the concept "real number" has much less analogy with the concept "cardinal number" than we, being misled by certain analogies, inclined to believe", that would have a good and honest sense. But just the opposite happens: one pretends to compare the "set" of real numbers in magnitude with that of cardinal numbers. The difference in kind between the two conceptions is represented, by a skew form of expression, as difference of extension. I believe, and I hope, that a future generation will laugh at this hocus pocus.