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The Book

  1. Nov 6, 2004 #1
    According to this paper,
    Paul Erdos (of Erdos number fame) conjectured the existence of The Book, a book that contains all the smallest proofs of mathematics arranged in lexical order. What are your thoughts on it, do you believe in the existence of such book?
  2. jcsd
  3. Nov 6, 2004 #2
    I do actually. :blushing:
  4. Nov 6, 2004 #3

    matt grime

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    Proofs from The Book, by Aigner and Ziegler (students of Erdos).

    Actually I don't think that Erdos said exactly that, though if you have an exact reference quoting him saying that I'll have to change my opnion. Size is not important in Erdos's opinion, it is elegance and beauty that counts, though he offered no formal definition of what constitutes a proof from the book, nor indeed a theorem that ought to have a proof in the book. Since it is relatively clear that "the space of all theorems" is not a set, with the naive definition of theorem and space, there is no hope of lexically ordering all proofs. Not even the axiom of choice makes any claims about ordering proper classes. Though I'm sure someone is about to shoot down that claim.
  5. Nov 6, 2004 #4
    I don't think Erdos ever said anything about the length of a proof or theorem, it was the most perfect, elegant, beautiful, etc proofs/theorems that made it into the book. example: Gauss' proof that the sum of the 1st 100 integers is 5050 is from the book (or Book, if you're a Platonist). So is Erdos' proof of the Prime Number theorem, but the original proof by those two French guys isn't a Book proof.

    I didn't know that the "space of theorems" isn't a set. Whatever it is, if we believe Erdos, God has them all listed together in 1 book and Erdos is probably reading it right now...
  6. Nov 8, 2004 #5

    matt grime

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    Well, accepting some large cardinal axioms we can easily create a "set" of theorems indexed by a proper class, eg one for each cardinal number, and that's just the tip of the iceberg. Of course all this is just playing around, and a proper proof theorist may object to this deliberately "classic" consrtuction, however a significant part of modern maths (ie category theory) often ignores whether things are sets or not.
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