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:::When I started this talk section, I had in mind the construction of rationals from integers, and I thought that algebraic numbers could be constructed from rationals in a similar way. The former is done by computing with pairs (''p'',''q'') ∈ ℤ×(ℤ\{0}) with the intended meaning ''p''/''q''; I thought the latter could be done by computing with polynomials, where one polynomial would denote one algebraic number, "viz. its root". Meanwhile I saw that even an irreducible polynomial has several roots, so that there can't be a one-to-one correspondence between polynomials and algebraic numbers, anyway. So I lost my original motivation for asking for irreducibility. Probably the proof is simplest in its current form; maybe a footenote could be added as you suggested.
:::In the enumeration table, I tried to distinguish several reasons for excluding a polynomial, a non-coprime set of coefficients being one of them, non-irreducibility being another one (admittely subsuming the former); when changing the table to produce duplicates these reasons would disappear, anyway. I used the gray parts to indicate (to myself, in the first place) the systematic way the polynomials are enumerated (nevertheless, I missed all polynomials containing ''x''<sup>3</sup> and ''x''<sup>4</sup>; see the new table; I hope it is complete now ...), but you are right: at least the exponent of "''x''<sup>1</sup>" isn't needed for that; I now deleted it. Concerning duplicates: should we have a reason "repetition" (or "duplicate"?) and not assign them a number; or should we assign them a number and mention somewhere that the enumeration is not bijective, but surjective, which suffices for countability? The former case would save some indentation space, since the ''x''<sup>4</sup> column could be immediately adjacent to the leftmost (number) column, as in each row at least one of them is empty. The latter case wouldn't save much, as "(-1 ± √5) / 2" (to be kept) is about as long as "repetition". - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 12:38, 16 December 2013 (UTC)
== Contrast 2nd theorem with sequence of rational numbers? ==
Cantor's 2nd theorem seems obvious at first glance to many people, as we usually are unable to imagine a sequence that could completely fill a whole interval. However, there are sequences (like that of all positive rational numbers) whose set of [[accumulation point]]s equals a whole interval (or even whole ℝ<sub>+</sub>; cf. the picture [[accumulation point|there]]). Mentioning this in the article might prevent novice readers from thinking "''Mathematicians make a big fuzz proving things that are obvious, anyway''", and might generally help to sharpen one's intuition about what a sequence ''can'' do in relation to an interval and what it ''cannot''. It would require, however, to explain the notion of an ''accumulation point'' (which is poorly represented in English Wikipedia in general). - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 11:52, 17 December 2013 (UTC)
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