Timeline for Why does this sum depend on the Axiom of Choice?
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
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| Mar 3, 2011 at 4:28 | comment | added | I. J. Kennedy | @Nate: Thanks, I see now that a misread the statement in the book. | |
| Apr 24, 2010 at 4:35 | vote | accept | I. J. Kennedy | ||
| Apr 23, 2010 at 16:53 | comment | added | Joel David Hamkins | Nate, I agree that this doesn't use AC. The convergence of a sequence can never depend on AC for the same reason that I give in my answer: if r is the sequence, then the statement "r converges" is absolute between the universe V and the relativized constructible universe L[r], where AC holds. So if you can prove that r converges in ZFC, then you can prove it in ZF. | |
| Apr 23, 2010 at 15:23 | comment | added | Nate Eldredge | After obtaining a copy of the book, the "theorem that was used" appears to be the theorem that a bounded increasing sequence of real numbers converges. (Grouping the terms as $(1 - \frac{1}{2}) + (\frac{1}{3} - \frac{1}{4}) + \dots$ gives an increasing sequence. Grouping as $1 + (-\frac{1}{2} + \frac{1}{3}) = \dots)$ shows the partial sums are bounded by 1.) But I'm not seeing how this theorem uses AC either. | |
| Apr 23, 2010 at 15:16 | history | edited | Charles Stewart |
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| Apr 23, 2010 at 2:20 | comment | added | Joel David Hamkins | Nate, yes, you don't need AC to prove that R is uncountable. | |
| Apr 23, 2010 at 2:14 | answer | added | Joel David Hamkins | timeline score: 21 | |
| Apr 23, 2010 at 2:13 | comment | added | Pete L. Clark | The statement in the book seems clearly false to me. On the other hand, the title is Mathematical Fallacies and Paradoxes, so caveat lector, I suppose. | |
| Apr 23, 2010 at 2:07 | comment | added | Nate Eldredge | It doesn't quite say that; it says that "the theorem that was used to show" that the series has a sum depends on the axiom of choice. It isn't clear from that passage what theorem they refer to. It may be possible to prove the fact without using that theorem. Later on that page, there is an allusion to some proof that the reals have greater cardinality than the integers that also made use of the axiom of choice, presumably a proof that was discussed earlier in the book. But unless I'm greatly mistaken, this fact doesn't rely on the axiom of choice either. | |
| Apr 23, 2010 at 2:04 | comment | added | Joel David Hamkins | The text says that "The theorem that was called upon to show that [this series] has a sum depends on the axiom of choice" and this is a little different than saying that the convergence of this particular series depends on AC. | |
| Apr 23, 2010 at 2:00 | comment | added | Ryan Thorngren | Perhaps the author means that one has to pick an arrangement out of the uncountably many possible arrangements. | |
| Apr 23, 2010 at 1:54 | history | asked | I. J. Kennedy | CC BY-SA 2.5 |