Timeline for Definability in HOD
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 5, 2013 at 15:14 | comment | added | Joel David Hamkins | The argument in this answer will apply even if you are allowed to quantify over $L[x]$, for any parameter or variable $x$. This includes ordinals and the transitive closure of $x$ and much more. But since the interpretation of $L[x]$ is absolute by forcing, again the truth of the assertion will be invariant by forcing. One could push this much harder, and consider statements quantifying over any definable class $K$ that is absolute by forcing. | |
| Dec 5, 2013 at 15:01 | comment | added | Joel David Hamkins | A set $x$ is in HOD when it is ordinal-definable in $V$, the larger ambient universe in which HOD is defined. That is, the definition runs in the larger universe $V$, not inside HOD. There is no expectation that the definition still works inside HOD, and as Andreas mentioned, generally the HOD as computed inside HOD might be strictly smaller than HOD. One can define HOD${}^n$ by iterating the HOD construction $n$ times. | |
| Dec 5, 2013 at 14:59 | comment | added | user38200 | I have a different question, it would be better to ask elsewhere maybe but here it goes: if $x$ is in HOD, then the defining formula for $x$ has quantifiers (bounded or unbounded) supposed to range over the whole universe, in the sense it is not restricted to HOD. If that's the case, then it would solve my problem already. | |
| Dec 5, 2013 at 14:55 | comment | added | Emil Jeřábek | I was about to write that after a bit more thought, the modification still does not work as it a priori restricts every quantifier to Ord or to $\bigcup\dots\bigcup x$ for a fixed number of $\bigcup$'s, and what one really needs is to be able to quantify over the transitive closure of $x$. However, Joel’s argument applies in this case as well. | |
| Dec 5, 2013 at 14:55 | vote | accept | user38200 | ||
| Dec 5, 2013 at 14:54 | comment | added | Joel David Hamkins | Did you mean bounded quantifiers? I think my argument does address that. | |
| Dec 5, 2013 at 14:53 | comment | added | user38200 | Could you address also the modification I included in the edit please? | |
| Dec 5, 2013 at 14:52 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |