Meta Math! The Quest for Omega · HackerPapers discussions

Meta Math! The Quest for Omega — mentioned by lioeters

Tue, 28 Jul 2026 01:47:04 GMT

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 real numbers will forever escape us, the notion of an arbitrary real has beautiful mathematical properties and is a concept that helps us to organize and understand the real world. Individual concepts in a theory do not need to have concrete meaning on their own; it is enough if the theory as a whole can be compared…

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Meta Math! The Quest for Omega — mentioned by floxy

Tue, 20 Feb 2024 18:49:50 GMT

"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 stunningly black high-mountain sky studded with pin-pricks of light." ...see chapter 5 of Meta Math! by Gregory Chaitin: https://arxiv.org/pdf/math/0404335.pdf

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Meta Math! The Quest for Omega — mentioned by floxy

Thu, 08 Dec 2022 23:24:52 GMT

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 line from 0 to 1 looks more and more like a Swiss cheese, more and more like a stunningly black high-mountain sky studded with pin-pricks of light."

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Meta Math! The Quest for Omega — mentioned by floxy

Tue, 27 Sep 2022 22:41:17 GMT

Maybe the GP is referring to something like Chapter 5 of Chaitin's Meta Math!. https://arxiv.org/pdf/math/0404335.pdf

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Meta Math! The Quest for Omega — mentioned by bentona

Wed, 20 May 2020 16:14:28 GMT

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

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Meta Math! The Quest for Omega — mentioned by GregBuchholz

Tue, 12 Jun 2018 16:57:12 GMT

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

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Meta Math! The Quest for Omega — mentioned by flubert

Mon, 09 Oct 2017 17:59:55 GMT

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).

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Meta Math! The Quest for Omega — mentioned by GregBuchholz

Thu, 22 Jun 2017 15:32:34 GMT

>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 meat of it). So constructivists don't believe in those types of numbers, and so get "choice" as a theorem instead of an axiom.

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Meta Math! The Quest for Omega — mentioned by surement

Mon, 10 Apr 2017 18:55:49 GMT

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

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Meta Math! The Quest for Omega — mentioned by GregBuchholz

Tue, 15 Nov 2016 19:06:06 GMT

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: Should You Believe? http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf Some interesting works by Gregory Chaitin Meta Math! The Quest for Omega https://arxiv.org/abs/math/0404335 Exploring Randomness https://www.cs.auckland.ac.nz/~chaitin/ait/ How Real are Real Numbers? https://arxiv.org/abs/math/0411418 People…

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Meta Math! The Quest for Omega — mentioned by GregBuchholz

Thu, 11 Aug 2016 19:45:06 GMT

>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 of theoretical physics to supply the missing parts; Bohr claimed that there are no missing parts. To most, their positions seemed diametrically opposed; however, if we can understand better what Bohr was trying to say, it is possible to reconcile their positions and believe them both. Each had an important truth to…

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Meta Math! The Quest for Omega — mentioned by GregBuchholz

Wed, 25 May 2016 17:01:00 GMT

>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 individually named or specified or even defined or referred to—constructively or not—within a formal language or within an individual FAS, has probability zero. Summary: reals are unnameable with probability one. So the set of real numbers, while natural—indeed, immediately given—geometrically, nevertheless remains quite…

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Meta Math! The Quest for Omega — mentioned by GregBuchholz

Thu, 28 Apr 2016 17:00:37 GMT

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

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Meta Math! The Quest for Omega — mentioned by deckar01

Mon, 27 Apr 2015 23:30:58 GMT

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 retrospect, because they are reduced into relationships between existing concepts. I would recommend trying all angles of attack when searching for a proof. If it starts to take too long or you hit a dead end, try a different angle. Even if you discover a proof, try your other opening moves, look for a more elegant…

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Meta Math! The Quest for Omega — mentioned by GregBuchholz

Tue, 29 Apr 2014 23:20:14 GMT

Can't mention Omega without linking to: Meta Math! http://arxiv.org/abs/math/0404335

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Meta Math! The Quest for Omega — mentioned by GregBuchholz

Mon, 28 Apr 2014 01:20:19 GMT

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, unknowable "real" numbers don't exist. http://arxiv.org/abs/math/0404335 ...(the impatient should jump to chapter 5). And maybe also: http://web.maths.unsw.edu.au/~norman/views2.htm

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Meta Math! The Quest for Omega — mentioned by andyjohnson0

Fri, 09 Aug 2013 09:02:11 GMT

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

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Meta Math! The Quest for Omega — mentioned by andyjohnson0

Tue, 06 Aug 2013 13:28:07 GMT

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] http://arxiv.org/abs/math/0404335

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Meta Math! The Quest for Omega — mentioned by adamgravitis

Sun, 07 Jul 2013 18:12:44 GMT

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 space, I'd love to chat with people familiar this stuff. Edit: arXiv PDF preprint is at http://arxiv.org/pdf/math/0404335.pdf

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Meta Math! The Quest for Omega — mentioned by jacobolus

Sat, 25 Sep 2010 04:28:45 GMT

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 scrolling through many pages of search results & running into journal paywalls): The only review at Amazon (2 stars): http://www.amazon.com/review/R1ZJVJX57T6YVA/ http://www.scientificexploration.org/journal/reviews/reviews... http://www.cs.auckland.ac.nz/CDMTCS/chaitin/siam.html A couple reviews whose "official" URLs are…

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