Timeline for What is the least $\alpha$ such that $L_\alpha$ contains a non-measurable set
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 23 at 21:18 | comment | added | Peter Gerdes | @JoelDavidHamkins Ahh, I was being an idiot because I got confused by the notation. I forgot that $L_\alpha$ isn't $V_\alpha$ for $L=V$. I take it that it's always $V_{\omega+2}$ that contains a non-measureable set even when $V=L$ | |
| Sep 13 at 18:40 | comment | added | Joel David Hamkins | No, because all sets of reals show up in $V_{\omega+2}$. The $L$ hierarchy is much slower than the $V$ hierarchy. | |
| Sep 13 at 18:33 | comment | added | Peter Gerdes | Is it also true that for any model V of ZFC the value of $\alpha$ such that $V_\alpha$ contains a non-measurable set is always equal to $\omega_1^V+1$? | |
| Sep 13 at 18:16 | comment | added | Peter Gerdes | @JoelDavidHamkins Also you correctly interpreted the question I was trying to ask. I think the other question can be made trivial since if you write the statement in terms of a set such that there is no sequence witnessing the set has a certain Lebesque measure and you identify then you can get failure at the first stage where you have an element of | |
| Sep 13 at 18:07 | vote | accept | Peter Gerdes | ||
| Sep 11 at 15:07 | comment | added | Andreas Blass | If $L_\gamma$ satisfies enough of ZFC, then the argument in this answer should work in $L_\gamma$ to show that a nonmeasurable-in-$L_\gamma$ set appears in $L_{{\omega_1}^{L_\gamma}}+1$. But what is "enough of ZFC"? You certainly need the existence of $\omega_1$ for the statement to make sense. I'd expect that to be equivalent, in models of the form $L_\gamma$, to the existence of $\mathcal P(\omega)$. Without that part of the power set axiom, you'd be looking for a nonmeasurable class of reals, which should appear at level $\gamma+1$ (just after the end of the universe). | |
| Sep 11 at 13:13 | comment | added | Joel David Hamkins | Yes, that is the question I answered. Your alternative question, also suggested by user 喻 良 on the main post, is subject to what else you want true in $L_\alpha$. For example, it will be sensitive to how exactly you define the measure and how you define what it means to be measurable. You should have a specific theory, such as $\text{ZFC}^-$. But in this case, once you specify the theory, the answer will be connected with the least $\gamma$ for which $L_\gamma$ satisfies that theory. For these reasons, I don't find that version of the question as robust. | |
| Sep 11 at 13:05 | comment | added | Gro-Tsen | Just to be clear, you answered the question “what is the smallest $α$ such that $L_α$ contains a subset of $2^ω$ which is not measurable in the universe ($V=L$)?”; another possible interpretation of the (to me ambiguous) question might have been “what is the smallest $α$ such that $L_α$ $\vDash$ «there exists a subset of $2^ω$ which is not measurable»?”. I don't know what can be said about the latter $α$, but it is obviously countable. | |
| Sep 11 at 11:30 | comment | added | Joel David Hamkins | @newaccount What would you mean by projective code? I guess just the (finite) projective formula, with its parameters. In our case here, we know the definition of a projective nonmeasurable set already (and my set above is projective in $L$, with no parameters), the code of it appears right away at a small finite stage. | |
| Sep 11 at 4:01 | comment | added | new account | When does the projective code (if that is a thing) for a nonmeasurable set appear? | |
| Sep 11 at 3:31 | comment | added | Elliot Glazer | Regarding the cases mentioned in the last paragraph: $(2^{\omega})^L$ (as well as any $\omega$-model of $\mathrm{RCA}_0$) is either measure 0, maximally nonmeasurable (inner measure 0 and outer measure 1), or equal to $2^{\omega},$ since it is closed under dyadic translation and from a real $r \not \in A=(2^{\omega})^L,$ we can construct infinitely many disjoint translations of $A.$ | |
| Sep 11 at 1:03 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
added 558 characters in body
|
| Sep 11 at 0:23 | history | answered | Joel David Hamkins | CC BY-SA 4.0 |