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[Edited so as to reflect answers and comments given so far]

Let $F$ be a real closed field. Then by Artin-Schreier theory, $F[i]$ is algebraically closed. If $F$ is further assumed to have cardinality at most continnum, then by a classical theorem of Steinitz [stating the isomorphism of any two uncountable algebraically closed fields of the same cardinality and characteristic], we can conclude that $F[i]$ is isomorphic to a subfield of $\Bbb{C}$ of complex numbers.

In particular, if $F$ is a non-archimedean real closed field of cardinality continuum, then $F[i]$ is isomorphic to $\Bbb{C}$ [The proof uses the axiom of choice in a serious way, by the way].

We can therefore conclude:

Theorem. Every nonarchimedean real closed field of power at most continuum is isomorphic to a subfield of $\Bbb{C}$.

As a special case, we may conclude that there is a subfield $F$ of $\Bbb{C}$ such that $F$ is a non-archimdean real closed field that, furthermore, has a subfield isomorphic to the field $\Bbb{R}$ of real numbers.

The above considerations allow me to state my questions.

Questions.

(a) [UNANSWERED] Is there an uncountable Borel non-Archimedean real closed field $F$ of $\Bbb{C}$?

NOTE: In his comment below Dave Marker asks whether this question has a negative answer if we further assume that $F[i]=\Bbb{C}$, then Gerald Edgar pointed out in his comment that this is indeed the case; based on a result that appears in a joint paper of his with Chris Miller.

(b) [ANSWERED] Is it possible for an uncountable such $F$ to be at least Lebesgue measurable ?*

NOTE. (b) has been answered. First Martin Goldstern pointed out that (b) follows from $MA + \lnot CH$; and that (b) is also true in any universe of set theory obtained by adding a Cohen real. Then Gerald Edgar pointed out that (b) is provable outright in $ZFC$ [see the answers below].

(c) [UNANSWERED] If the answer to (a) is positive, does the answer change if we insist for $F$ to have a subfield isomorphic to $\Bbb{R}$.

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Can we rule out a) if we also ask that ${\mathbb C}=F[i]$? –  Dave Marker May 28 '11 at 3:01
    
Dave: good question; I will add it to the list of questions. –  Ali Enayat May 28 '11 at 18:22
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A Borel subfield $F$ of $\mathbb C$ which is equal neither to $\mathbb C$ nor to $\mathbb R$ must have Hausdorff dimension $0$. And then, of course, $F+iF$ also has Hausdorff dimension $0$. ams.org/journal-getitem?pii=S0002-9939-02-06653-4 –  Gerald Edgar May 28 '11 at 20:29
    
@Gerald--Thanks I had forgotten your result with Chris –  Dave Marker May 28 '11 at 21:59
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If $F$ is a Borel subfield of $\mathbb C$ (or more generally any ``Borel field") that is isomorphic to the reals. Then, as Ali points out, the ordering is Borel. There is a real coding the embedding $q\mapsto a_q$ of the rationals into $F$. The isomorphism between the reals and $F$ is given by $f(x)= y$ if $x<q \Leftrightarrow y<a_q$ for all rationals, which has a Borel graph. –  Dave Marker Jun 2 '11 at 12:52
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2 Answers

A partial answer to (b): Consistently, yes. Löwenheim-Skolem implies that there are non-Archimedean real closed fields of cardinality $\aleph_1$, and it is consistent that all sets of size $\aleph_1$ are of measure zero. (For example, this follows from MA plus non-CH.)

Alternatively: Take any model $V$ of set theory, let $K\in V$ be an uncountable non-Archimedean real closed field, and add a Cohen real $c$ to $V$. In $V[c]$, the set of old reals has now measure zero, and $K$ is still a non-Archimedean real closed field.

I have a feeling that an absoluteness argument should now help to get the existence of $K$ in $V$, but I cannot get it to work. I do not even know if I can get a definable (say: analytic) $K$ in $V[c]$ (though it seems to me that I can get a $K$ containing a perfect set).

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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. –  Gerald Edgar May 27 '11 at 13:44
    
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 $ZFC$. –  Ali Enayat May 27 '11 at 14:17
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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? –  Ali Enayat May 27 '11 at 14:25
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Is there a Cantor set $K$ in $\mathbb R$ of algebraically independent elements?

Thinning it out, if necessary, we can assume $K^n$ has Hausdorff dimension $0$ for all $n$, and there are still other things algebraically independent of it. Then $F_1 = \mathbb Q(K)$ is at least an analytic subfield, still of dimension zero, and its algebraic completion $F_2$ is again analytic, now isomorphic to $\mathbb C$, but not all of $\mathbb C$ and not equal to $\mathbb R$. So $F_2$ is still of Hausdorff dimension $0$. Of course it has a subfield $F_3$ isomorphic to $\mathbb R$. (Probably many different ones?) So if we now add something algebraically independent of that, specifying that it is infinitely large, say, compared to $F_3$, then can't we now do a real-completion to get something still analytic and nonarchimedean real closed?

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Edgar: there certainly is a Cantor set &K$ in $\BBb{R}$ of algebraically independent elements (so far, so good). But let me ponder the rest of your argument. –  Ali Enayat May 28 '11 at 21:58
    
@Gerald-- There are perfect sets of algebraically independent reals. (For a logical argument--a perfect set of mutually generic Cohen reals has this property and the existence of a perfect tree of algebraically independent reals is absolute. This can probably be translated into a reasonably easy Baire Category proof). One difficulty I see with your sketch is that it is not clear you can find $F_3$ as Borel subfield of $F_2$. –  Dave Marker May 28 '11 at 22:02
    
... and that last real-completion maybe doesn't make sense, either. In any case, though, as noted in the original question, if $F$ is any real closed field of power c, then $F[i]$ is isomorphic to $\mathbb C$, and my argument here says there is a Lebesgue null subfield of $\mathbb C$ isomorphic to $\mathbb C$, and thus inside it a Lebesgue null copy of $F$. Answering (b) with no need for axioms beyond ZFC. But probably not Borel or even analytic. –  Gerald Edgar May 29 '11 at 0:16
    
Edgar and Dave: it appears to me that the null copy of $F$ can be arranged to be analytic since there is an infinite binary tree all of whose branches are algebraically independent (by a fine-tuning of the argument sketched by Dave). I will edit the question to reflect this development. –  Ali Enayat May 29 '11 at 0:51
    
I would like to see more details of the $F$ analytic claim –  Gerald Edgar May 29 '11 at 12:55
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