arXiv:math/0404335math.HO
This book presents a personal account of the mathematics and metamathematics of the 20th century leading up to the discovery of the halting probability Omega. The emphasis is on history of ideas and philosophical implications.
Linked by 10 identifiable commenters in 20 comments across 20 threads, from Sep 2010 to Jul 2026.
"Why should I believe in a real number if I can’t calculate it, if I can’t prove what its bits are, and if I can’t even refer to it? And each of these things happens with probability one! The real line from 0 to 1 looks more and more like a Swiss cheese, more and more like a…
You may, or may not enjoy chapter 5: https://arxiv.org/pdf/math/0404335.pdf ..."Why should I believe in a real number if I can’t calculate it, if I can’t prove what its bits are, and if I can’t even refer to it? And each of these things happens with probability one! The real…
Maybe the GP is referring to something like Chapter 5 of Chaitin's Meta Math!. https://arxiv.org/pdf/math/0404335.pdf
This is by Gregory Chaitin, who also published a pretty interesting & related book called "META MATH!" [1] I especially like the Turing / Godel equivalency stuff he discusses. [1] https://arxiv.org/pdf/math/0404335.pdf
A good book to read to get aquainted with these notions is "Meta Math" by Gregory Chaitin. The impatient can jump to chapter 5. https://arxiv.org/abs/math/0404335
If you want to seed doubts in your mind about the real numbers, you can get that at: https://arxiv.org/abs/math/0404335 ...as well (skip to chapter 5 for the impatient).
>Constructive mathematicians believe the reals exists. I think when people mention the "reals" they are usually referring to the uncountable reals, which you can't construct, compute, name, or know: https://arxiv.org/abs/math/0404335 ...(maybe skip to chapter 5 to get to the…
Also the author of Meta Math! The Quest for Omega [0], where this topic is discussed at length. [0] https://arxiv.org/abs/math/0404335
For anyone who might be misled, here are some links that people can use to judge for themselves how "crank" someone is: Professor Norman Wildberger's homepage: http://web.maths.unsw.edu.au/~norman/ ...a paper by Wildberger tackling some of the issues in this thread: Set Theory…
>and onto irrelevant metaphysical tangents. For those with some time to spare, you might enjoy: Clearing Up Mysteries -- The Original Goal http://bayes.wustl.edu/etj/articles/cmystery.pdf "Put most briefly, Einstein held that the QM formalism is incomplete and that it is the job…
>How do these finitists handle things like the real numbers? You may enjoy "Meta Math" by Gregory Chaitin (jump to Chapter 5 for the impatient): http://arxiv.org/abs/math/0404335 "Finally, and perhaps even more devastatingly, it turns out that the set of all reals that can be…
You could always be a intuitionist/constructivist, and enjoy "choice" as a theorem. It seems completely reasonable to give up uncountable infinities. http://arxiv.org/abs/math/0404335
I'm not sure "try not to use [proof by contradiction] unless you really need it" is the optimal conclusion. In "Meta Math"[1], Gregory Chaitin explores the question: "What makes a mathematical concept desirable?" Simplicity was the solution. The best theorems seem obvious in…
Can't mention Omega without linking to: Meta Math! http://arxiv.org/abs/math/0404335
Of course that argument only applies when talking about the standard real number system. http://arxiv.org/pdf/1007.3018.pdf?origin=publication_detail http://arxiv.org/pdf/0811.0164.pdf ...of course you can make a pretty good case that uncountable, uncomputable, unnameable…
Definitely. I'd recommend his book Meta Math: The Quest for Omega [1]. He posted the full text of the book on arXiv [2]. [1] https://www.amazon.co.uk/Meta-Maths-Gregory-J-Chaitin/dp/184... [2] http://arxiv.org/abs/math/0404335
For a deep look at Omega (Chaitin's constant) I'd recommend Chaitin's book Meta Math: The Quest for Omega [1]. A fascinating book. Chaitin has also uploaded the full text (pdf) of the book to Arxiv.org [2]. [1] http://www.amazon.co.uk/Meta-Maths-Gregory-J-Chaitin/dp/1843... [2]…
Anyone interested in this would immensely enjoy Meta Math! by Gregory Chaitin. Basically, he demonstrates that there are theorems which are randomly true: they're true for no particular reason. It's, of course, deeply connected with Gödel's work. If anyone's working in this…
Chaitin’s website about the book, which seems to include the full contents (in HMTL format): http://www.cs.auckland.ac.nz/~chaitin/lm.html A PDF draft from 1994 at Arxiv: http://arxiv.org/abs/chao-dyn/9407009 Some other links about the book (putting these here to save people…
Connected through shared HN threads, not inferred similarity or a recommendation.
From page 12: > Why should we believe in real numbers, if most of them, it turns out,[^15] are maximally unknowable like Ω? [^16] The footnotes: > [^15]: See the chapter entitled The Labyrinth of the Continuum in [Chaitin, 2005] > [^16]: In spite of the fact that most individual…
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