[FOM] Artin-Schreier for Q, without choice?

[Noah Schweber]

Shoenfield absoluteness provides a great "hammer" for getting rid of
choice. One's favorite construction of the algebraic numbers yields a
particular (indeed, computable) countable structure A which can be coded
canonically by a single real. By Shoenfield, no $\Sigma^1_2$ fact about it
can depend on choice. In particular, "A is real-closed" is an *arithmetic*
(in the computability-theoretic sense) property of A.

The key point here is that any usual A is explicitly countable. Indeed,
suppose W is a model of ZF and in W the real r codes an algebraic
completion of Q. Then all $\Sigma^1_2$ facts about r are absolute between W
and the inner model L[W], and the latter satisfies choice. "Exotic"
structures like Lauchli's avoid this by being not representable in this way
(and indeed not even well-orderable).

Of course, this doesn't address applications of LEM (which Ingo
Blechschmidt's response does), so this is a limited approach. But it's
still quite nice.



 - Noah

On Sun, Jan 5, 2020 at 12:22 AM Timothy Y. Chow <tchow at math.princeton.edu>
wrote:

> In ZF, one cannot prove that the algebraic closure of Q (the rationals) is
> unique.  In fact, Lauchli has constructed an algebraic closure of Q which
> has no real-closed subfield.
>
> But now suppose we just want to build *an* algebraic closure L of Q with
> all the properties we know and love, and don't care that there might exist
> "exotic" algebraic closures of Q that are not isomorphic to L.  Since N
> (the natural numbers) is given to us with a well ordering, it is
> straightforward to construct L without needing to invoke AC.
>
> What about the Artin-Schreier theorem?  It seems straightforward to
> construct (a particular version of) the real algebraic numbers, but what
> about proving that they form a real-closed field?  Glancing at a proof of
> Artin-Schreier, I instinctively feel that AC is not needed for the
> specific example of Q, but in light of Lauchli's result, I am worried that
> I am missing some subtle point.  Also, in the absence of AC, do we need to
> be cautious about the precise statement of any of the usual properties of
> real-closed fields---such as those listed on Wikipedia?
>
> https://en.wikipedia.org/wiki/Real_closed_field
>
> Tim
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