
Hilbert's Grand JavaScript School (2015 Edition) - jessaustin
http://raganwald.com/2015/04/24/hilberts-school.html
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zfghjk
Yeah, an infinity cubed is countable. Just iterate over the infinite number of
finite diagonals, like you did for the square (a diagonal is any set of points
where the coordinates sum to a particular constant).

Similarly, an infinity to the 4th power is countable, as is an infinity to the
n'th power for any finite n. Interesting question - is an infinity to an
infinitieth power still countable?

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AnkhMorporkian
Surely it must be by induction. Since n+1 is always countable, n+2 must
similarly be.

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wetmore
It is not. The induction you speak of only works to prove for finite values.
For example, let A be some countable set; with induction you could show that
A^n is countable for any particular n = 1, 2, ...

However, this doesn't work for the product of A with itself countably-many
times. Your induction never "reaches" infinity; it only shows that it works
for any finite number n. Sure, there are infinitely-many such n, but every
single natural number is finite.

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AnkhMorporkian
I'm not a math wizard, but everything I was taught in all of my calculus
classes made me believe that infinity isn't an actual tangible concept outside
of the alephs, and only applicable in limits, which surely follows the rules
of induction. Am I missing something here?

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ColinWright
Induction suffices to show that (oo)^n is countable for every _n_ a natural
number, but that does not suffice to show that (oo)^(oo) is countable, because
the power is not a natural number. The induction you have only proves the step
that if X^n is countable then X^(n+1) is countable. That never proves the case
X^(oo)

There is a thing called transfinite induction, and there you have to show the
separate case that if predicate P holds for all n_i then it holds for the
limit. As stated elsewhere you can't do that in this case because it's not
true. 2^(oo) is uncountable.

    
    
      > Am I missing something here?
    

Yes. You are forming perfectly reasonable conjectures based on limited
experience, but those conjectures turn out to be false when you study the
subject in its own right, instead of just those bits you need for the limited
version of calculus that you've done. It's a problem I often have when
teaching people this stuff that they have only ever dealt with "nice"
functions and "easy" situations. I wrote a little about that here:

[http://www.solipsys.co.uk/new/PokingTheDustyCorners.html?HN_...](http://www.solipsys.co.uk/new/PokingTheDustyCorners.html?HN_20150427)

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AnkhMorporkian
I really appreciate that explanation and link. Thanks so much!

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javajosh
Is it just me, or does this JavaScript style fill you with dread? JS already
has about 1000 ways to do things, and with new syntax that's going up a few
orders of magnitude....

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dbpokorny
Why not use scheme streams and the SICP chapter 4 query language? This was
done in the 1980s.

[http://mitpress.mit.edu/sicp/full-text/book/book-
Z-H-24.html...](http://mitpress.mit.edu/sicp/full-text/book/book-
Z-H-24.html#%_sec_3.5.2)

3.5.2 Infinite Streams

