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Comment by cstross | Hacker News Reader

Rolls Eyes

Someone here hasn't heard of Georg Cantor.

(More to the point, this is only a paradox if you assume that there is only one transfinite number. A brisk tour of "Infinity and the Mind" by Rudy Rucker sufficed to disabuse me of that idea some years ago. Time for a re-read, I think.)

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That's either a nice troll or you don't know what you think you know!

Possibly. What do you think I don't know?

Interesting question. The problem is that you've mentioned there being more than one transfinite number, when this question is actually about convergence. It appears that you've simply gone "Alleged paradox - infinity - must be ignorant about Cantor and the uncountables."

In fact, Cantor and the uncountables have nothing to do with it. If you think otherwise then I'd be interested in seeing a more complete explanation of your comment.

Well, there's a transfinite implicit in the series of exponentiations -- we have aleph-0 exponentiations, don't we? And if we knock off that first one, we've got (aleph-0 minus 1) exponentiations.

(NB: I don't understand your use of the term "convergence" -- must be either something I've forgotten in the third of a century since I studied maths, or something above the level I reached. (Sub A-level.) (See also: Dunning-Kruger effect.))

Technically, f(x) converges to y as x -> infinity if for every epsilon there is an X such that |f(x) - y| < epsilon for all x > X ("the error becomes arbitrarily small"). For instance, 1/x converges to 0 as x -> infinity (pick X = 1/epsilon), but the sequence 0, 1, 0, 1, 0, ... does not converge to any y (pick epsilon < 1/2 to see this).

In particular, the sequence r, r^r, r^r^r, ..., where r^2 = 2, does converge (to 2; the first terms are approximately 1.41, 1.63, 1.76, ...); the sequence 2, 2^2, 2^2^2, ..., however, does not (the first terms are 2, 4, 16, ... - and it only grows faster from there.)

Convergence is what happens when you have a sequence of numbers, and they get closer and closer to some fixed number. Some sequences converge and some don't, and part of what's going on with this "paradox" is that one of the sequences it implicitly asks you to consider doesn't converge.

Yes, there's a transfinite number implicit in the sequence of exponentiations, namely aleph-0. (It might be better to say omega-0 -- an infinite ordinal rather than an infinite cardinal -- but that's a technicality that doesn't really matter here.)

But the problem with the "proof" doesn't have anything to do with the existence of infinite cardinals larger than aleph-0.

It's 4am here and I need some sleep - choose 2 numbers from within the convergent range and do the same slight of hand with them. Clearly we can argue that x^^\inf = 4 has no convergence and so makes no sense but what if the equations are both in the converging region x € [e^−e, e^1/e] about [0.066, 1.445]:

  {1}    x^^\inf = 2

we've shown in the OP that x^2 = 2 is a solution.

Consider,

  {2}    x^^\inf = e^(2/e)

similarly to the OP we do

  {3}    x^ (x^^\inf) = x^ (e^(2/e))

but this is just

  {4}    x^^\inf =  x^ (e^(2/e))

and substituting on the left from {2}

  {5}    e^(2/e) = x^ (e^(2/e))

gives us (I think)

  {6}    x = e^ (2e^ (-1-(2/e))) ; ~~1.42267

Hence we've shown that

  {7}    2 ~~ 1.423

Which isn't as neat as 2==4 but equally troublesome.

There is no issue now with convergence is there (or there shouldn't be if I chose my numbers correctly)?

---

Notations

  ^^ : ascii version of Knuths up-arrow notation
  \inf : first countable infinity, aleph-0 
  € : in the range, it's really a Euro symbol, but close enough
  ~~ : approximately equal to

===

Solution

  y = x^2 
    = 9 give solution (3,9)
  y = x^2
    = 4 gives solution (2,4)

9=4?

They're just different points on the same curve ... doh! We're not showing that 9=4 we're showing that (3,9) and (2,4) are point solutions of y = x^2.

In the same way we have above 2 solution of y = x^^\inf.

That this doesn't have much at all to do with there being multiple transfinite numbers.

No infinity other than Aleph-naught appears to be involved in this. Why do you think the existence of other aleph numbers matters here?

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