- #26

Hurkyl

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.010101010... (1/99) is not in the list exactly as Cantor's function real number result is not in the list.

Then why are you claiming:

My list is a decimal representation of any rational number in Cantor's first argument specific list.

When there is clearly at least one rational number whose decimal representation is not in your list?

This is the exact state between Cnator's function some result and the R list.

But there's a very important logical piece you're missing. Cantor's second diagonal argument is applied to

**EVERY**list of real numbers, and that's how you conclude |N|<|R|.

However, you are applying the diagonal argument to a single list, and if it is the case that no rational number appears twice on the list, all we can conclude is |N|<=|Q|.

That is the key you are missing. In order to prove |S|<|T| for sets S and T, you have to prove that

**EVERY FUNCTION**from S to T is missing an element of t. Proving it for only one function doesn't cut it

^{1}.

^{1}: except when there exists only one function from S to T, in which case proving it for one function coincides with proving it for every function. This situation can only happen when S or T is the empty set, or when |S|=|T|=1.

Therefore there are no limits to the number of ways that Q list can be rearranged to give us some new Q number (which is not in the list) as a Cantor's function result.

Sure there is. There are at most |N|^|N| =

*c*ways.

And I bet I can give you a list of rational numbers for which "Cantor's function" does not give you a rational number for any rearrangement.

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