Timeline for What are the hidden assumptions behind Harvey Friedman's claim, CSR?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| May 16, 2022 at 18:50 | comment | added | Noah Schweber | @Corbin It doesn't matter that they don't stay (and AoC plays no role here). For example: pick the first element that lands in the first "level-1" Cauchy box $B_1$; now pick the first element after that one that lands in the first "level-2" Cauchy subbox $B_2$ of $B_1$; now pick the first element after that one that lands in the first "level-3" Cauchy subbox $B_3$ of $B_2$; etc. This process is totally choice-free due to the "pick the first" bit, and is quickly convergent. I think you might be thinking of "subsequence" in the more restrictive sense of just throwing away an initial segment? | |
| May 16, 2022 at 18:45 | comment | added | Corbin | I understand what Peter and Noah are saying. I think that my issue must be with selecting an infinite subsequence. The terms of $q_i$ do visit every Cauchy bucket, but they do not stay; arithmetic subsequences of $q_i$ drift in and out of buckets. So I'm not exactly sure how to select an infinite subsequence without AoC. But this is classical set theory, so AoC is fine. (And I'm voting to close.) | |
| May 16, 2022 at 18:39 | comment | added | Corbin | @NoahSchweber: I have no problem with this being moved to MSE; I thought MO was the proper place for archeological queries, but I don't really care either way. | |
| May 16, 2022 at 17:30 | comment | added | Noah Schweber | To clarify my previous comment: CSR doesn't assert that there is a unique such subsequence - in any sense. Having the initial sequence be "randomly spread out" just means it has lots of such rapidly-convergent subsequences, converging to lots of different things. (Separately, I think this would be more appropriate for MSE.) | |
| May 15, 2022 at 18:15 | comment | added | Noah Schweber | Visiting every Cauchy bucket makes it easier to find convergent subsequences. | |
| May 15, 2022 at 16:56 | review | Close votes | |||
| May 24, 2022 at 3:05 | |||||
| May 15, 2022 at 16:12 | comment | added | Peter LeFanu Lumsdaine | I don’t follow why you think $(q_i)$ should be a counterexample to CSR. Here’s the simple classical proof outline for CSR: (1) any infinite sequence in $[0,1]$ has a convergent subsequence; then (2) for any specified modulus of converegence/Cauchyness, any Cauchy sequence has a subsequence converging at the specified rate. It seems clear how to apply these steps to your $(q_i)$? | |
| May 15, 2022 at 15:48 | history | asked | Corbin | CC BY-SA 4.0 |