Comments on: The hydra game http://math.andrej.com/2008/02/02/the-hydra-game/ Mathematics for computers Mon, 30 Jan 2012 07:14:48 +0000 hourly 1 http://wordpress.org/?v=3.2-beta2-18055 By: Andrej Bauer http://math.andrej.com/2008/02/02/the-hydra-game/comment-page-1/#comment-16556 Andrej Bauer Wed, 27 Jul 2011 06:02:32 +0000 http://math.andrej.com/2008/02/02/the-hydra-game/#comment-16556 @Bo: I think it's ok. The head itself gets a 0, but because it is growing out of a lower node, we turn that into $\omega^0 = 1$. So, the only way to get $0$ is to have just one head without any necks, in other words the root. For example, if we had one head growing out of the root we would assign $0$ to the head, and then $\omega^0 = 1$ to the whole thing. @Bo: I think it’s ok. The head itself gets a 0, but because it is growing out of a lower node, we turn that into $\omega^0 = 1$. So, the only way to get $0$ is to have just one head without any necks, in other words the root. For example, if we had one head growing out of the root we would assign $0$ to the head, and then $\omega^0 = 1$ to the whole thing.

]]> By: Bo http://math.andrej.com/2008/02/02/the-hydra-game/comment-page-1/#comment-16554 Bo Tue, 26 Jul 2011 21:27:20 +0000 http://math.andrej.com/2008/02/02/the-hydra-game/#comment-16554 You said a head gets the number 0 but in your examples is seems to be 1. You said a head gets the number 0 but in your examples is seems to be 1.

]]> By: “To infinity and beyond: The struggle to save arithmetic “ « A kind of library http://math.andrej.com/2008/02/02/the-hydra-game/comment-page-1/#comment-13692 “To infinity and beyond: The struggle to save arithmetic “ « A kind of library Mon, 23 Aug 2010 23:01:02 +0000 http://math.andrej.com/2008/02/02/the-hydra-game/#comment-13692 [...] In the late 70s a series of sentences such that Peano arithmetic cannot prove and cannot prove were discovered. They have in common that, unlike Goedel's examples, they are not coding logical notions. Instead, they are genuine mathematical statements of the kind that a person working in combinatorics would find interesting. The example I gave is Goodstein's theorem (an undergraduate student of mine at Caltech wrote a nice paper on this a few years ago, "The termite and the tower."). There are others. A nice one is about a game, Hercules and the Hydra. [...] [...] In the late 70s a series of sentences such that Peano arithmetic cannot prove and cannot prove were discovered. They have in common that, unlike Goedel’s examples, they are not coding logical notions. Instead, they are genuine mathematical statements of the kind that a person working in combinatorics would find interesting. The example I gave is Goodstein’s theorem (an undergraduate student of mine at Caltech wrote a nice paper on this a few years ago, “The termite and the tower.”). There are others. A nice one is about a game, Hercules and the Hydra. [...]

]]> By: Xamuel http://math.andrej.com/2008/02/02/the-hydra-game/comment-page-1/#comment-13518 Xamuel Fri, 18 Jun 2010 21:14:34 +0000 http://math.andrej.com/2008/02/02/the-hydra-game/#comment-13518 Hey Andrej, I just wrote up another similar application of ordinal arithmetic: a method of surfing the internet without getting infinitely sidetracked following links. Kind of like this hydra, except with webpages and links instead of hydra-heads. Here it is: http://www.xamuel.com/internet-surfing-ordinals/ Hey Andrej,

I just wrote up another similar application of ordinal arithmetic: a method of surfing the internet without getting infinitely sidetracked following links. Kind of like this hydra, except with webpages and links instead of hydra-heads. Here it is: http://www.xamuel.com/internet-surfing-ordinals/

]]> By: Andrej Bauer http://math.andrej.com/2008/02/02/the-hydra-game/comment-page-1/#comment-13489 Andrej Bauer Mon, 31 May 2010 04:54:20 +0000 http://math.andrej.com/2008/02/02/the-hydra-game/#comment-13489 @Peter: I see them fine on my Firefox, what platform are you using? I fixed the problem with multiplication by 4 from the wrong side, thank you for spotting it. @Peter: I see them fine on my Firefox, what platform are you using? I fixed the problem with multiplication by 4 from the wrong side, thank you for spotting it.

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