Kindle Notes & Highlights
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March 2 - March 3, 2019
Thought shall be harder, heart the keener, Courage the greater, as our might lessens
He made out a new distinction between consistent and inconsistent collections: only the former were sets; the latter (such as the collection of all sets or of all the ordinals) were not.
His leap upward from towering heights to bring back knowledge of the lower structure has characterized subsequent work in set theory, a century now and more after that moment.
In what passes for the real world, no one could make sense of Cantor’s proof. Long after, Zermelo said of it that “the intuition of time is applied here to a process that goes beyond all intuition, and a fictitious entity is posited of which it is assumed that it could make successive arbitrary choices”—over a span longer than time’s.
And now it wasn’t just that certain collections were simply too large to be consistent: in 1901 Bertrand Russell showed that set theory generated paradoxes that hadn’t anything to do with alephs or ordinals at all. He had found that legitimate ways of defining a set (via the properties shared by its members) led to nonsense.
Sets were so intuitively clear when we began that we were happy to reduce the mysteries of number to them. Now they have uncontrollably swollen and multiplied, and which are consistent, which inconsistent, and for whom? Their very nature has grown incoherent.
Dedekind said that for him a set was a closed bag with specific things in it which you couldn’t see and knew nothing about, except that they were distinct and really there. A few minutes passed. Cantor, immensely tall, flung out his arm toward the wild landscape: “A set,” he said, “I think of as an abyss.”
The question of infinity had brought mathematics to the edge of uncertainty. —Joseph Warren Dauben
His great work on set theory of 1883 was prefaced with three quotations, the last of which was, “The time will come when these things which are now hidden from you will be brought into the light.”
It was precisely Cantor’s daring diagonal which Kurt Gödel turned round to prove that there were more true propositions than proofs: that in any sufficiently rich formal system there would be statements which could neither be proved nor refuted.
Gödel’s own later work, and that of the American logician Paul Cohen, then showed that the Continuum Hypothesis was one of these statements.
Mathematics is permanent revolution. Gödel’s inevitably followed from the radical mathematics invented by Cantor.
Cantor’s transfinite arithmetic is this Unconformity on a universal scale. It has disrupted our sedate understanding of the mind and its world, and from its fracture a new understanding has yet fully to emerge. When it does—when the doors of our perception are finally cleansed, as William Blake promised—then everything will appear as it is: infinite. But which infinity will we see?
