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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: (This text was in the section "Second theorem", just after the paragraph starting with "The proof is complete since ...")
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''<sub>''N''</sub>, ''b''<sub>''N''</sub>) or [''a''<sub>∞</sub>, ''b''<sub>∞</sub>], respectively. However, in case 2 where ''a''<sub>∞</sub> = ''b''<sub>∞</sub>, there is no interval from which an arbitrary algebraic number may be chosen. The value of ''a''<sub>∞</sub> must not be algebraic, because asserting that it is algebraic leads to a contradiction with the first theorem.
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