Hilbert's Grand JavaScript School (2015 Edition)
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Hilbert's Grand JavaScript School (2015 Edition)
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Re: Hilbert's Grand JavaScript School (2015 Edition)
#2Yeah, 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?
Re: Hilbert's Grand JavaScript School (2015 Edition)
#3Yeah, 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?
Surely it must be by induction. Since n+1 is always countable, n+2 must similarly be.
Re: Hilbert's Grand JavaScript School (2015 Edition)
#4Yeah, 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?
"is an infinity to an infinitieth power still countable?"
The countable product of even finite sets is not, in general, countable. For example, we can identify a real number with a binary sequence; in particular we can form a bijection between the set of all real numbers and a subset of the space of infinite binary sequences. If the product of the set {0,1} with itself countably-many times was countable, there would be countably-many real numbers, which we know is not true.
Re: Hilbert's Grand JavaScript School (2015 Edition)
#5Yeah, 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?
Surely it must be by induction. Since n+1 is always countable, n+2 must similarly be.
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.
Re: Hilbert's Grand JavaScript School (2015 Edition)
#6Earlier quoted context omitted.
Surely it must be by induction. Since n+1 is always countable, n+2 must similarly be.
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,…
Right. A simple example to show that induction does not work to infinity:
0 Re: Hilbert's Grand JavaScript School (2015 Edition)
#7Yeah, 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?
> is an infinity to an infinitieth power still countable?
It shouldn't be. You can use the same principle as showing that reals aren't countable:
Suppose you have an ordered (infinite) set of the infinite-dimensional vectors. Doesn't matter the ordering.
Then construct another vector by the following: the Nth dimension of the vector is different from the Nth dimension of the Nth element of the set. Could be +1, could be -1, could be whatever, long as it's different.
Note that this doesn't actually require infinity^infinity, it's just {any set of >1 element}^infinity.
The new vector is, by definition, different from every element of the set, as the Nth dimension's value is different than the Nth element of the set's Nth value, for all N. Hence the new vector isn't in the set.
(Alternatively, you can biject {0-9}^infinity with the reals between 0 and 1. And the reals between 0 and 1 aren't countable.)
Re: Hilbert's Grand JavaScript School (2015 Edition)
#8Is 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....
Re: Hilbert's Grand JavaScript School (2015 Edition)
#9Yeah, 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?
It looks like Dr. Hilbert “Bertie” David reads Hacker News:
Re: Hilbert's Grand JavaScript School (2015 Edition)
#10Why 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...
3.5.2 Infinite Streams