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sniffnoyHere's a link.
For those of you that are just tuning in: Let's define ||n|| to be the complexity of n, by which in this context we mean the smallest number of 1s needed to write n using any combination of addition and multiplication. (Note that this is the number 1, not the decimal digit 1. Allowing you to write "11" and say you've only used two 1s would just be silly.)
Then we can define the defect of n, denoted δ(n), to be the quantity ||n||-3log3n; and we can then consider the set of all defects δ(n), as n ranges over all natural numbers. Surprisingly, this set is well-ordered, and its order type is ωω.
...OK, this won't be surprising to you if you've spoken to me anytime in, say, the past several years. But it should be a surprise to almost everyone else. And it's pretty damned neat regardless.
Thanks are of course due to Joshua Zelinsky -- who, after all, defined the object of study of this paper in the first place (defect, that is, not complexity) -- and to Juan Arias de Reyna who not only helped a lot with the editing but also helped organize several of the ideas in the paper in the first place. And other people, but, well, you can check the acknowledgements if you really care about that.
We'll see where I can get this published. In the meantime, this should be quite a bit more readable than the old draft sitting around on my website.
Now I guess it's on to the next paper (for now)... or rather, it already has been for a while...
For those of you that are just tuning in: Let's define ||n|| to be the complexity of n, by which in this context we mean the smallest number of 1s needed to write n using any combination of addition and multiplication. (Note that this is the number 1, not the decimal digit 1. Allowing you to write "11" and say you've only used two 1s would just be silly.)
Then we can define the defect of n, denoted δ(n), to be the quantity ||n||-3log3n; and we can then consider the set of all defects δ(n), as n ranges over all natural numbers. Surprisingly, this set is well-ordered, and its order type is ωω.
...OK, this won't be surprising to you if you've spoken to me anytime in, say, the past several years. But it should be a surprise to almost everyone else. And it's pretty damned neat regardless.
Thanks are of course due to Joshua Zelinsky -- who, after all, defined the object of study of this paper in the first place (defect, that is, not complexity) -- and to Juan Arias de Reyna who not only helped a lot with the editing but also helped organize several of the ideas in the paper in the first place. And other people, but, well, you can check the acknowledgements if you really care about that.
We'll see where I can get this published. In the meantime, this should be quite a bit more readable than the old draft sitting around on my website.
Now I guess it's on to the next paper (for now)... or rather, it already has been for a while...
