Talk:Cantor's first set theory article: Difference between revisions

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{{Did you know nominations/Georg Cantor's first set theory article}}

== Reply to "Why doesn't Cantor's second theorem apply to a world of purely algebraic numbers?" ==

Ipsic asks the following: Explain what seems to me to be a subtle point. Why doesn't Cantor's second theorem apply to a world of purely algebraic numbers?

Here's the text that came with this request:

Note, that under cases 1 and 3, above, the real number in [''a'', ''b''] that is not a contained in the sequence may be chosen to be any of an infinite number of algebraic numbers that are contained within the intervals (''a''_''N'', ''b''_''N'') or [''a''_∞, ''b''_∞], respectively. However, in case 2 where ''a''_∞ = ''b''_∞, there is no interval from which an arbitrary algebraic number may be chosen. The value of ''a''_∞ must not be algebraic, because asserting that it is algebraic leads to a contradiction with the first theorem.

I agree that this can be a subtle point. The important point is to distinguish what Cantor's theorem does in general from what it does in a particular application, such as being applied to the sequence of all algebraic numbers in an interval. In general, Cantor's theorem only guarantees that the real number obtained from his construction is not in the given sequence. In many applications, such as applying it to the sequence of rationals in (0, 1) as in the section "Example of Cantor's construction", you are not guaranteed to obtain a transcendental number but in this case you are guaranteed to obtain an number that is not rational. Applying the theorem to this case, you obtain <math>\sqrt{2}-1</math>, which is a non-rational algebraic number.

However, if you apply Cantor's theorem to the sequence of all algebraic numbers in an interval (''a'',&nbsp;''b''), you are guaranteed to obtain non-algebraic number in the interval.

Since you brought up that you can hit an algebraic number in Case 1 or Case 3, I wish to point out that the section "Dense sequences" proves that a dense sequence—such as, the sequence of algebraic numbers in an interval—never ends up in Case 1 or Case 3, so your argument fails here. To obtain a transcendental number, Cantor is using such a dense sequence.

If you still find this confusing, please let me know. By the way, if you have questions about an article, the questions really belong on the Talk pages and not in the article's text. --[[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 02:00, 13 September 2018 (UTC)