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Thread: Conway chained arrow notation

  1. #1 Conway chained arrow notation 
    Moderator Moderator AlexP's Avatar
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    In Conway chained arrow notation, I understand chains of three, for example: 2 -> 3 -> 3 = 2^^^3 (using Knuth up-arrow notation), but I haven't been able to understand how chains of four work. I got from Wikipedia that 3 -> 2 -> 2 -> 2 = 3 -> 3 -> 8, but I don't understand how exactly. What does the fourth number do to the rest of the chain? Can any of you smart math people (serpicojr) enlighten me?


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  3. #2  
    Forum Professor serpicojr's Avatar
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    Well, I'm not familiar with this notation, so I don't have any intuitive notion of what adding a third arrow/fourth number does (except "it makes the number really damn big"). However, I can justify the calculation for you using the axioms. Recall that the axioms are:

    1. p->q = p<sup>q</sup> for positive integers p, q
    2. X->1 = X for any chain X
    3. X->p->(q+1) = X->(X->(...(X->(X)->q)...)->q)->q, where this new chain has p X's, p-1 q's, and p-1 sets of paretheses, and where X is any chain, p and q are any positive integers

    So to "calculate" 3->2->2->2, we do the following:

    1. Let X = 3->2, so that 3->2->2->2 = X->2->2. Then rule 3 says X->2->2 = X->2->(1+1) = X->(X)->1 = 3->2->(3->2)->1.

    2. Rule 1 says that 3->2 = 3<sup>2</sup> = 9, so 3->2->(3->2)->1 = 3->2->9->1.

    3. Rule 2 says we can knock off 1 from the end of any chain, so 3->2->9->1 = 3->2->9.

    4. Rule 3 says 3->2->9 = 3->2->(8+1) = 3->(3)->8 = 3->3->8.

    Really, this just looks like an efficient way of writing a certain set of big numbers. I imagine that it's useful in the following senses:

    a. It's probably a natural operation that arises out of complexity or combinatorial arguments. (E.g., you might be describing some iterative process, and the number of steps it takes to complete is somehow expressed well by these numbers.)

    b. It's probably not difficult to come up with asymptotics (i.e., estimates) for these numbers; if it is, then it's at least probably not difficult to compare two of these numbers. (E.g., so we could verify that one process is faster than another.)

    I doubt that there are any useful intuitive notions of what chains of length four or five or ten or a billion really "mean". In fact, I think it's best to think of it this way: you understand what chains of length one, two, and three mean; you understand the basic differences in size between these and that, as you add each arrow, the numbers become much larger; and the jump from two to three arrows is much bigger than the jump from one to two arrows; so, for each arrow we add, the numbers became absurdly larger, and the difference is more and more absurd the more arrows we add.


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  4. #3  
    Moderator Moderator AlexP's Avatar
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    serpicojr, that's exactly what I was hoping to avoid learning. So, thank you, I've been convinced that I shouldn't try to take the easy way out and should just learn it. Thanks.
    "There is a kind of lazy pleasure in useless and out-of-the-way erudition." -Jorge Luis Borges
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