Timeline for "Lexicographic" ordering on ${\cal P}(\omega)$
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 13, 2016 at 19:30 | comment | added | Emil Jeřábek | Now that I think about it, one can show directly without any density argument that R embeds in the order: fix an enumeration of Q by natural numbers, and map each real to the set of codes of rationals that precede it. | |
| Apr 13, 2016 at 18:55 | comment | added | Joel David Hamkins | You are right, and my argument is wrong. I'll edit later. {}<{0} and there is nothing between. | |
| Apr 13, 2016 at 16:54 | comment | added | Emil Jeřábek | @Joel: I may misunderstand the definition, but it seems to me that N < N-{1} < N-{0}. You even claim yourself in another comment that the order is dense on cofinite sets, which these two sets are. | |
| Apr 13, 2016 at 14:29 | vote | accept | Dominic van der Zypen | ||
| Apr 13, 2016 at 14:19 | comment | added | Nate Eldredge | If I'm not mistaken, you can embed your order into Baire space $\mathbb{Z}^\omega$, with its lexicographic order, by listing the elements of a set in increasing order and padding out finite sets with -1. And I'm pretty sure you can't embed $\omega_1$ into Baire space. According to this, you can embed the lex order on Baire space into the reals with their usual order (specifically the irrationals) and you definitely can't embed $\omega_1$ in the reals. | |
| Apr 13, 2016 at 14:19 | answer | added | Joel David Hamkins | timeline score: 6 | |
| Apr 13, 2016 at 14:14 | comment | added | Joel David Hamkins | Emil, it isn't dense on the infinite sets, since $\mathbb{N}<\mathbb{N}-\{0\}$, but there is nothing in between. But it is dense on the finite sets. | |
| Apr 13, 2016 at 13:47 | comment | added | Emil Jeřábek | The restriction of this order to infinite sets is a dense order, hence it contains a copy of Q, hence it contains a copy of any countable linear order. | |
| Apr 13, 2016 at 13:31 | history | edited | Dominic van der Zypen | CC BY-SA 3.0 |
Changed definition of $<$
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| Apr 13, 2016 at 13:23 | history | asked | Dominic van der Zypen | CC BY-SA 3.0 |