Timeline for Nonstandard Reals in the Complex Plane
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 27, 2011 at 14:25 | comment | added | Ali Enayat | Martin: thanks for your observation about the consistency of (b). Here is a thought: in the presence of sufficiently large cardinals, Woodin's $\Sigma^2_{1}$-absoluteness theorem, when coupled with your answer, assures us (it seems) that the answer to (b) is positive in forcing extensions satisfying $CH$. So perhaps already in $ZFC+CH$ the answer to (b) is positive? | |
| May 27, 2011 at 14:17 | comment | added | Ali Enayat | Gerald: precisely because of the point you made about countable non-archimedean real closed fields, my questions all are formulted for uncountable real closed fields. The most interesting case is when such a field $F$ contains $Bbb{R}$ as a subfield [part (c) of the question], but it seems to me that even arranging for $F$ to be uncountable and measurable is nontrivial in $ZFC$. | |
| May 27, 2011 at 13:44 | comment | added | Gerald Edgar | Why bother with $\aleph_1$? There are non-archimedean real closed fields of cardinality $\aleph_0$, and certainly when (non-constructively?) embedded into $\mathbb C$ they are Borel sets. Of course the interesting question would be the case where the real closed field extends $\mathbb R$ and thus has cardinal of the continuum. | |
| May 27, 2011 at 7:47 | history | answered | Goldstern | CC BY-SA 3.0 |