Timeline for Existence of $\alpha$-sequence of infinitesimal sequences with $\alpha>\omega_1$
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Apr 23, 2019 at 11:47 | vote | accept | ar.grig | ||
| Apr 23, 2019 at 11:17 | comment | added | ar.grig | @EmilJeřábek: if the answer is negative then there exists α-sequence with $\alpha>\omega_1$ | |
| Apr 23, 2019 at 11:05 | history | edited | ar.grig | CC BY-SA 4.0 |
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| Apr 23, 2019 at 10:59 | comment | added | ar.grig | @Wojowu: infinitesimal $\omega$-sequence is the sequence with countably many members which converges to zero. | |
| Apr 23, 2019 at 10:35 | review | Close votes | |||
| Apr 23, 2019 at 18:22 | |||||
| Apr 23, 2019 at 10:34 | comment | added | Asaf Karagila♦ | Do you know about Hausdorff gaps? | |
| Apr 23, 2019 at 10:12 | answer | added | Wojowu | timeline score: 2 | |
| Apr 23, 2019 at 9:58 | comment | added | Johannes Schürz | I believe infinitesimal sequence refers to the field-extension of $\mathbb{R}$ adding non-standard reals $r$ , i.e. $\forall q \in \mathbb{Q}^+ \colon 0< r<q$. The way you do this, you consider $\mathbb{Q}^\omega$ modulo eventually <. | |
| Apr 23, 2019 at 9:51 | comment | added | Emil Jeřábek | A straightforward diagonalization argument shows that if $f$ is any sequence of rationals, and $G$ is a countable set of rational sequences such that $f\ll g$ for each $g\in G$ (meaning $\lim_n f(n)/g(n)=0$), then there exists a rational sequence $h$ such that $f\ll h$ and $h\ll g$ for all $g\in G$. Thus, for any fixed $f$, one can construct by induction a sequence $\{g_\alpha:\alpha<\omega_1\}$ such that $\alpha<\beta\implies f\ll g_\beta\ll g_\alpha$. This is likely a counterexample to the question, except that I don’t know what exactly “infinitesimal sequence” means. | |
| Apr 23, 2019 at 9:39 | answer | added | Johannes Schürz | timeline score: 1 | |
| Apr 23, 2019 at 9:33 | comment | added | Emil Jeřábek | Is the title a typo? The question seems to only speak about $\alpha<\omega_1$. | |
| Apr 23, 2019 at 9:27 | comment | added | Wojowu | What is an infinitesimal $\omega$-sequence? | |
| Apr 23, 2019 at 8:53 | history | asked | ar.grig | CC BY-SA 4.0 |