Wolfram Science Group
Phoenix, AZ USA
Registered: Aug 2003
Reals, or infinite sequences of arbitrary integers, are not enumerable. Yes if the length of the sequence (or in your version the number of dimensions) is bounded, then you can indeed put them into one to one correspondance with the natural numbers. But not when it is infinite. This was shown in the famous diagonalization argument of Cantor back in the 19th century.
A standard account in set theory terms is here -
The proof works by contradiction - you assume you've got 'em all, and then ask about one of them, and find you can't have got it yet. Say I've enumerated the sequences, any way you please, giving me a look up table with the natural number on the left (the n-th series) and the series on the right.
n1 <-> a1 a2 a3 a4 a5 ...
n2 <-> b1 b2 b3 b4 b5 ...
n3 <-> c1 c2 c3 c4 c5 ...
n4 <-> d1 d2 d3 d4 d5 ...
n5 <-> e1 e2 e3 e4 e5 ...
Now consider the sequence X == a1+1, b2+1, c3+1, d4+1, e5+1... Starting from the diagonal of your enumeration, taking one term from each series you've numbered. But then changing that term.
Is X in the enumeration? Well, it isn't n1, because it disagrees with n1 in the first term of the series. It isn't n2, because it disagrees with n2 at the second term of the series. It isn't n3, because it disagrees with n3 in the third term of the series - etc. Ergo, X is not in the enumeration.
But we did not even need to specify what the enumeration was, so this conclusion follows regardless of what enumeration method we use. All we did was assume that we had them all, and then found we had missed at least one. We got a contradiction from the assumption that we could get them all in a 1 to 1 correspondance with the natural numbers. Ergo, we can't get them all.
To relate this intuitively to what you are trying, notice that down at terms eight hundred gazillion, I might indeed have a candidate n-800 gazillion which did match my sequence X for the first 800 gazillion digits. But that is not enough. It has to match in the 800 gazillion-th and 1st digit, too.
Truly infinite in sequence length, or in number of digits, or in number of dimensions, is truly different from anything bounded in that respect. You can get the one infinity of higher and higher integers. And you can get it in more and more (but finite, bounded) dimensions or terms.
You can number all the integer location points in a space with 800 gazillion dimensions. More, you can number all the rational location points - a set that is what we call "dense" in the reals - in a space with 800 gazillion dimensions. But 800 gazillion is not infinity, and you can't number all the integer points in a space with truly infinite dimensions.
Multiple infinities are tricky. I hope this helps.
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