6 ms·

Graham's Number used to be the biggest number I could name, but now I understand that TREE(3) ( https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem https://e

Graham's Number used to be the biggest number I could name, but now I understand that TREE(3) ( https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem ) is bigger still by a truly incredible amount (and TREE(1) = 1, and TREE(2) = 3). Unfortunately I don't understand that Wiki page. The page links to https://en.wikipedia.org/wiki/Friedman%27s_SSCG_function https://en.wikipedia.org/wiki/Friedman%27s_SSCG_function , which is supposedly even bigger but I can't follow the text of the first page so won't attempt to read that.

bionoid 9y ago
I will take this opportunity to promote the YouTube channels of Brady Haran[0], Numberphile specifically. It has all sorts of interesting videos, including Graham's number [1] (one of the videos featuring Ron Graham) and TREE(3) [2]. His other channels are worth checking out too, Sixty Symbols and Deep Sky Videos being my favorites. To be clear, I have no affiliation with this, just a fan. [0] http://bradyharan.com/ http://bradyharan.com/ [1] https://www.youtube.com/watch?v=HX8bihEe3nA&list=PLt5AfwLFPxWKZEG7KVg6HfdN2uWFLIB5q https://www.youtube.com/watch?v=HX8bihEe3nA&list=PLt5AfwLFPx... [2] https://www.youtube.com/watch?v=3P6DWAwwViU&t=54s https://www.youtube.com/watch?v=3P6DWAwwViU&t=54s
- Balgair 9y ago
  Numberphile is one of those series that makes me like math again. That strange, crazy, extreme logic is just great. Then I try to work with it, and I am made glad that I didn't pursue that career. I especially like the intro to Graham's number and the rough proof that it is unthinkably large, that your head would literally implode into a black hole an uncountable amount of times over. Math is so boggling. Numbers that are literally super-natural exist and are 'useful' at the same time, yet are impossible to mortals all the same. It's this weird end-around we manage with math/logic. That 'floor falling out from underneath you' feeling, a happy giddiness and smallness. Sorry, Graham's Number does something fierce to my head, it seems.
speakeron 9y ago
TREE(3) is indistinguishable from zero compared to Loader's number[1], which basically takes every bit pattern up to some n and expresses this as a program in the Calculus of Constructions. This system is a bit weaker than being Turing complete, but the programs do always terminate (this makes the number computable compared with, say, the Busy Beaver number which does a similar thing with Turing complete programs). It also has the geek cred of being represented by an obfuscated C program (the unobfuscated verson is also available[2]). [1] http://googology.wikia.com/wiki/Loader%27s_number http://googology.wikia.com/wiki/Loader%27s_number [2] https://github.com/rcls/busy https://github.com/rcls/busy
- Retric 9y ago
  I can trivially define a function that's vastly larger than that. What's interesting is when these numbers are useful for something. k(1) = TREE(1), K(2) = TREE(TREE(2)), K(3) = TREE(TREE( <(repeats (TREE(3) time (TREE(3))) Sequence 1, 2, non comprehensibly large number. Now define m(x) in terms of k and m(x-1).
  - speakeron 9y ago
    > k(1) = TREE(1), K(2) = TREE(TREE(2)), K(3) = TREE(TREE( <(repeats (TREE(3) time (TREE(3))) What you've made here is what's known in googology circles as a 'salad number'. It doesn't make a conceptual leap above what the TREE function can do anyway and is kind of similar to saying TREE(3) + 1. SCG(13) is much bigger than your K(3) and Loader's number is much bigger than SCG(13). There are (computable) numbers bigger than Loader's number which are mentioned in the googology link in my previous post (Greedy clicque sequences and finite promise games).
    - Retric 9y ago
      Yea, the point was to pick something trivial. K(3) is larger than TREE( any obvious function to a non mathematician of any printable number). But it's the m(x) that's actually large in that example.
semi-extrinsic 9y ago
That SSCG function is pretty astonishing: > SSCG(3) is not only larger than TREE(3), it is much, much larger than TREE(TREE(…TREE(3)…)) where the total nesting depth of the formula is TREE(3) levels of the TREE function. notwithstanding that TREE(3) is pretty crazy in itself: > an extremely weak lower bound for TREE(3), is A(A(...A(1)...)), where the number of A's is A(187196), and A() is a version of Ackermann's function. Graham's number, for example, is approximately A^64(4)

bionoid 9y ago

I will take this opportunity to promote the YouTube channels of Brady Haran[0], Numberphile specifically. It has all sorts of interesting videos, including Graham's number [1] (one of the videos featuring Ron Graham) and TREE(3) [2]. His other channels are worth checking out too, Sixty Symbols and Deep Sky Videos being my favorites. To be clear, I have no affiliation with this, just a fan. [0] http://bradyharan.com/ http://bradyharan.com/ [1] https://www.youtube.com/watch?v=HX8bihEe3nA&list=PLt5AfwLFPxWKZEG7KVg6HfdN2uWFLIB5q https://www.youtube.com/watch?v=HX8bihEe3nA&list=PLt5AfwLFPx... [2] https://www.youtube.com/watch?v=3P6DWAwwViU&t=54s https://www.youtube.com/watch?v=3P6DWAwwViU&t=54s

Balgair 9y ago
Numberphile is one of those series that makes me like math again. That strange, crazy, extreme logic is just great. Then I try to work with it, and I am made glad that I didn't pursue that career. I especially like the intro to Graham's number and the rough proof that it is unthinkably large, that your head would literally implode into a black hole an uncountable amount of times over. Math is so boggling. Numbers that are literally super-natural exist and are 'useful' at the same time, yet are impossible to mortals all the same. It's this weird end-around we manage with math/logic. That 'floor falling out from underneath you' feeling, a happy giddiness and smallness. Sorry, Graham's Number does something fierce to my head, it seems.

speakeron 9y ago

TREE(3) is indistinguishable from zero compared to Loader's number[1], which basically takes every bit pattern up to some n and expresses this as a program in the Calculus of Constructions. This system is a bit weaker than being Turing complete, but the programs do always terminate (this makes the number computable compared with, say, the Busy Beaver number which does a similar thing with Turing complete programs). It also has the geek cred of being represented by an obfuscated C program (the unobfuscated verson is also available[2]). [1] http://googology.wikia.com/wiki/Loader%27s_number http://googology.wikia.com/wiki/Loader%27s_number [2] https://github.com/rcls/busy https://github.com/rcls/busy

Retric 9y ago
I can trivially define a function that's vastly larger than that. What's interesting is when these numbers are useful for something. k(1) = TREE(1), K(2) = TREE(TREE(2)), K(3) = TREE(TREE( <(repeats (TREE(3) time (TREE(3))) Sequence 1, 2, non comprehensibly large number. Now define m(x) in terms of k and m(x-1).
- speakeron 9y ago
  > k(1) = TREE(1), K(2) = TREE(TREE(2)), K(3) = TREE(TREE( <(repeats (TREE(3) time (TREE(3))) What you've made here is what's known in googology circles as a 'salad number'. It doesn't make a conceptual leap above what the TREE function can do anyway and is kind of similar to saying TREE(3) + 1. SCG(13) is much bigger than your K(3) and Loader's number is much bigger than SCG(13). There are (computable) numbers bigger than Loader's number which are mentioned in the googology link in my previous post (Greedy clicque sequences and finite promise games).
  - Retric 9y ago
    Yea, the point was to pick something trivial. K(3) is larger than TREE( any obvious function to a non mathematician of any printable number). But it's the m(x) that's actually large in that example.

semi-extrinsic 9y ago

That SSCG function is pretty astonishing: > SSCG(3) is not only larger than TREE(3), it is much, much larger than TREE(TREE(…TREE(3)…)) where the total nesting depth of the formula is TREE(3) levels of the TREE function. notwithstanding that TREE(3) is pretty crazy in itself: > an extremely weak lower bound for TREE(3), is A(A(...A(1)...)), where the number of A's is A(187196), and A() is a version of Ackermann's function. Graham's number, for example, is approximately A^64(4)