Timeline for Is the theory of ordinals in Cantor normal form with just addition decidable?
Current License: CC BY-SA 4.0
11 events
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| Aug 27 at 19:55 | vote | accept | cody | ||
| Aug 27 at 18:16 | history | edited | Noah Schweber |
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| Aug 27 at 13:20 | history | became hot network question | |||
| Aug 27 at 12:52 | answer | added | Emil Jeřábek | timeline score: 27 | |
| Aug 27 at 7:45 | comment | added | Emil Jeřábek | The theory of ordinals with $+$ is decidable. I do not remember off-hand what happens with $\omega^-$, but anyway, you should look at Ehrenfeucht's paper first. | |
| Aug 27 at 2:18 | history | edited | Alex Kruckman |
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| Aug 26 at 22:09 | history | edited | cody | CC BY-SA 4.0 |
Fix notation to more standard one.
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| Aug 26 at 22:08 | comment | added | cody | Both fair comments. I'll probably keep the title for continuity though (and I wouldn't mind an answer to the other question as well). | |
| Aug 26 at 20:46 | comment | added | Joel David Hamkins | Regarding the title, Cantor normal form applies to all ordinals, not just ordinals up to $\varepsilon_0$. Every ordinal can be uniquely expressed as a finite sum $\omega^{\alpha_n}+\cdots+\omega^{\alpha_0}$ with $\alpha_n\geq\cdots\geq\alpha_0$. Below $\varepsilon_0$, the interesting thing is that you can get a nice terminating finite hereditary representation, with the exponents also represented this way. At $\varepsilon_0$, however, the hereditary representation starts not to work, since the representation $\varepsilon_0=\omega^{\omega^{\omega^{\cdot^{\cdot}}}}$ doesn't terminate. | |
| Aug 26 at 20:35 | comment | added | Joel David Hamkins | If you intend to ask about the structure whose domain is the ordinals below $\varepsilon_0$, then you should write $(\varepsilon_0,\ldots)$, since $\text{Ord}^{<\varepsilon_0}$ would usually denote the proper class of all sequences of (arbitrary) ordinals, with length less than $\varepsilon_0$. Similarly, $\omega$ is the set of finite ordinals, whereas $\text{Ord}^{<\omega}$ is the class of all finite sequences of ordinals. | |
| Aug 26 at 19:40 | history | asked | cody | CC BY-SA 4.0 |