Interpreting the projective hierarchy in expansions of the real line

Hieronymi, Philipp; Tychonievich, Michael

doi:10.1090/S0002-9939-2014-12023-5

Interpreting the projective hierarchy in expansions of the real line
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by Philipp Hieronymi and Michael Tychonievich PDF

Proc. Amer. Math. Soc. 142 (2014), 3259-3267 Request permission

Abstract:

We give a criterion when an expansion of the ordered set of real numbers defines the image of $(\mathbb {R},+,\cdot ,\mathbb {N})$ under a semialgebraic injection. In particular, we show that for a non-quadratic irrational number $\alpha$, the expansion of the ordered $\mathbb {Q}(\alpha )$-vector space of real numbers by $\mathbb {N}$ defines multiplication on $\mathbb {R}$.

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