Of nested and coiled serpents in logic
[ by Charles Cameron — another exploration into forms of insight — in this case the Matrioshka effect, spiral staircases and the like, with a glance at holy winds and human fingertips ]
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Oh look! The Weekly Standard did a blog about the HuffPost blog on my blog item, http://t.co/1ixOTgZhly Original: http://t.co/ODrj0MG4Rv
— Carol Rosenberg (@carolrosenberg) August 23, 2013
A mention of blogs about blogs about blogs seems to me to qualify for the “nested serpent” category of forms that are worth watching out for, the nest (or spiral, from which I am guessing the nest is not entirely separable) being of particular interest because while seemingly simple enough, it all too often reaches at one end or both into the infinities, where paradox meets epiphany… as my second example will show.
But first, by sheer good fortune, I came across this verse from the book of Ecclesiastes as I was polishing this post for publication:
The winde goeth toward the South, and turneth about vnto the North; it whirleth about continually, and the winde returneth againe according to his circuits.
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That’s the pattern we’re looking for, and I ran across it recently in a comment my friend Allen Stairs made, and the response he received:
Imagine a Russian doll set isomorphic to the natural numbers. Bet that didn't work very well, did it?
— Allen Stairs (@AllenStairs) August 24, 2013
@haroldpollack Equipollent, perhaps…
— Allen Stairs (@AllenStairs) August 24, 2013
Okay, I didn’t follow — so I asked Allen for an explanation, and he wrote me:
Actually. “equipollent” was a bad choice of terms. “Equinumerous” wold have been better.
But the thing about numbers and those dolls: natural numbers have their identity intrinsically, so to speak. In set theory, one way to represent them is as the series
1 = {Ø}, 2 = {Ø,{Ø}}, 3 = {Ø,{Ø,{Ø}}}, etc.
In fact, we can even use the simpler construction
1 = {Ø}, 2 = {{Ø}}, 3 = {{{Ø}}}, etc.
So if we’re given the set, its structure tells us which number it is. I
Now a finite set of Russian dolls does much the same thing. We could count the innermost one as 1, the next as 2, the next as 3, and so on, and if you were given the doll, you’d be able to tell which number it represented. Or if we wanted, we could let the outermost doll represent 1, and work our way in. But if we take the set of all natural numbers, things get a little wonkier. The thing about a set of dolls is that there’s an outer one; the charm is in the fact that there’s a place to start opening them. So suppose we have an infinitely nested set of dolls. What number does the outermost one correspond to? It can’t be a natural number, because for any natural number, the nesting would have to be finite. It can’t be the infinite number Aleph-null because among other things, if the nesting is infinite downward then each doll has the same structure as the one that encloses it, and so it seems that there’s no way for the individual dolls to represent distinct integers.
Now if we’re given the whole set of dolls, there’s a sort of substitute: match dolls to numbers depending on how many “predecessors” they have. The outermost doll has no predecessors, so let it be 1; the next one in has 1 predecessor, so let it be 2. And so on. But we still have a problem: there’s nothing about the doll itself that tells us which integer it represents.
So my little point was that Harold’s joke was about “how many?” but the thing about the dolls is that they might seem at first to have the right structure to represent the natural numbers, and yet they don’t — at least, not the whole set of natural numbers.
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Did Ecclesiastes mention the winds? Here’s a discussion of wind spirals from David Avram‘s The Spell of the Sensuous: Perception and Language in a More-Than-Human World:
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