Timeline for Definability in HOD
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 5, 2013 at 14:55 | vote | accept | user38200 | ||
| Dec 5, 2013 at 14:52 | answer | added | Joel David Hamkins | timeline score: 12 | |
| Dec 5, 2013 at 14:46 | history | edited | user38200 | CC BY-SA 3.0 |
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| Dec 5, 2013 at 14:45 | comment | added | Emil Jeřábek | The modification only shifts the problem one level down, that is, $x$ and $x'$ are indistinguishable by such a formula if $\forall y\in x\,\exists y'\in x'\,(y\cap\mathrm{Ord}=y'\cap\mathrm{Ord})$ and vice versa. Couldn’t you allow the formula to contain arbitrary bounded quantifiers? | |
| Dec 5, 2013 at 14:44 | comment | added | Andreas Blass | The correction should also affect the third paragraph. Presumably, the definitions $\psi$ that you want are definitions in the universe, not in $HOD$. | |
| Dec 5, 2013 at 14:39 | history | edited | user38200 | CC BY-SA 3.0 |
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| Dec 5, 2013 at 14:38 | comment | added | Andreas Blass | In the second paragraph you wrote, about an object $x\in HOD$, that "$x$ is hereditarily definable from ordinals in $HOD$." If "in $HOD$" was intended to apply to "definable", then it is wrong; a set in $HOD$ need not be $OD$ in $HOD$. If "in $HOD$" was intended to apply to "ordinals", then it is trivial, as all ordinals are in $HOD$. | |
| Dec 5, 2013 at 14:31 | history | edited | user38200 | CC BY-SA 3.0 |
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| Dec 5, 2013 at 14:25 | comment | added | Emil Jeřábek | If this is true for $x$ not an ordinal itself, $\psi$ can only (usefully) access $x$ through subformulas of the form $y\in x$, where $y$ is an ordinal (quantified or parameter). Thus, if $x\ne x'$ are in $\mathrm{HOD}-\mathrm{Ord}$ and $x\cap\mathrm{Ord}=x'\cap\mathrm{Ord}$, they cannot both have a defining formula of this form. | |
| Dec 5, 2013 at 14:14 | history | asked | user38200 | CC BY-SA 3.0 |