Timeline for Splitting infinite sets
Current License: CC BY-SA 2.5
11 events
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Apr 29, 2011 at 0:34 | vote | accept | Andrés E. Caicedo | ||
| Jan 9, 2011 at 7:11 | answer | added | Eric Hall | timeline score: 9 | |
| Jan 6, 2011 at 8:04 | comment | added | Andrés E. Caicedo | @Michael: Finally, there are yet other two problems. One, the compactness theorem itself requires a non-trivial amount of choice to hold, so we would need to prove a suitable version of compactness as part of our approach, which seems more ambitious. Two, we do not want to conclude from our approach that ${\mathbb N}$ injects into $X$, so the compactness argument we would require would have to be unlike any I can think of at the moment. | |
| Jan 6, 2011 at 8:01 | comment | added | Andrés E. Caicedo | @Michael: Given $X$, the other possibility I can think of would be to have the language contain a relational symbol $R$ and a function symbol $f$, and set up a theory that would have a model of the form $X\sqcup Y$ where $R$ is interpreted by $X$, $Y$ is infinite, and $f\upharpoonright X$ is a function from $X$ onto $Y$ with the preimage of each element of $Y$ being infinite. Again, the lack of an appropriate version of Löwenheim–Skolem is a serious issue here. | |
| Jan 6, 2011 at 7:58 | comment | added | Andrés E. Caicedo | @Michael: I am not sure I see how. Given $X$ you would need to devise a theory (presumably with infinitely many relational symbols that would play the role of the partition) for which there is a model with universe equipotent to $X$. This seems very delicate (we do not have the Löwenheim–Skolem theorem without choice). Moreover, we need to ensure that the theory has no models of size $Y$ for any amorphous $Y$. Plus, we would need to anticipate (to set up the language!) the infinite set $Z$ such that $X$ is a union indexed by $Z$ of infinite sets (I think $Z$ doesn't need to be countable). | |
| Jan 6, 2011 at 7:47 | comment | added | Michael Hardy | Could one of the compactness theorems of logic be used here? | |
| Jan 5, 2011 at 2:53 | comment | added | Andrés E. Caicedo | @Ricky: Sure. That was poorly phrased. I've changed the sentence into what I really meant. | |
| Jan 5, 2011 at 2:52 | history | edited | Andrés E. Caicedo | CC BY-SA 2.5 |
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| Jan 5, 2011 at 2:18 | comment | added | user5810 | If X is infinite, then 2^X can be mapped onto omega. This is provable in ZF and has nothing to do with X being Dedekind finite. | |
| Jan 5, 2011 at 1:11 | history | asked | Andrés E. Caicedo | CC BY-SA 2.5 |