Timeline for Upper bound on natural ordinal sum
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Jul 16 at 16:03 | vote | accept | opfromthestart | ||
| Jul 16 at 15:53 | comment | added | Emil Jeřábek | As noted in comments to the other (now deleted) answer, $\alpha\oplus\beta\le\alpha+\beta+\alpha$ is false in general: it fails e.g. for $\alpha=\omega$, $\beta=\omega^2+1$. | |
| Jul 16 at 15:44 | comment | added | opfromthestart | Would expressions like $a\oplus b < a+b+a$ fall under the same condition? My context is that I am writing a computer verified proof, and I need specific bounds on what $(a\oplus b)-a-b$ can be. | |
| Jul 16 at 13:04 | comment | added | Emil Jeřábek | Hmm, with $\cdot$, there are exceptions if $\alpha$ or $\beta$ is $1$: we have $\alpha\oplus\beta\le\alpha+\beta^2$, which is smaller than the bound above if $\beta=1\le\alpha$. But I think the optimality should hold for $\alpha,\beta\ge2$. | |
| Jul 16 at 12:50 | comment | added | Emil Jeřábek | I’ll refrain from further edits. But the bound can be shown optimal in that if $t(\alpha,\beta)$ is any expression using $\alpha$, $\beta$, $+$, $\min$, $\max$, and arbitrary ordinal constants, such that $t(\alpha,\beta)\ge\alpha\oplus\beta$ for all $\alpha,\beta$, then $t(\alpha,\beta)\ge\min\{\alpha+\beta+\beta,\beta+\alpha+\alpha\}$ for all $\alpha,\beta$. This may well be true even for expressions using $\cdot$, but this might require much more work to check. | |
| Jul 16 at 10:45 | history | edited | Emil Jeřábek | CC BY-SA 4.0 |
The last bound is, in fact, better (or no worse) than the other in all cases, thus simplify.
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| Jul 16 at 10:00 | history | edited | Emil Jeřábek | CC BY-SA 4.0 |
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| Jul 16 at 9:27 | history | edited | Emil Jeřábek | CC BY-SA 4.0 |
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| Jul 16 at 9:02 | history | edited | Emil Jeřábek | CC BY-SA 4.0 |
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| Jul 16 at 8:56 | history | answered | Emil Jeřábek | CC BY-SA 4.0 |