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Is there an infinitary formula $\Phi$ (in the language of fields in which countable conjunctions and disjunctions are permitted with finitely many quantifiers $L_{\omega_1,\omega}$) which characetrizes the field of real numbers up to isomorphism of ordered fields (in a single model of set theory or in the real world)?

If yes, does $\Phi$ respects different models of set theory, i.e. in every model $\mathcal{M}$ of set theory the formula $\Phi^\mathcal{M}$ defines the field (isomorphic (in $\mathcal{M}$ to) $\mathbb{R}^\mathcal{M}$?

user48900
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  • What do you mean an infinitary formula? Is it a formula in $L_{\kappa\lambda}$ for some $\kappa,\lambda$? In that case, do you have further assumptions on the models (i.e. both containing $\kappa$ and $\lambda$ and agree that they are cardinals, etc.)? – Asaf Karagila Dec 15 '13 at 16:30
  • I edited above. But if there is no such formula in $L_{\omega_1,\omega}$, we can allow a formula in $L_{\kappa,\lambda}$. – user48900 Dec 15 '13 at 16:32
  • There is a problem with your question, now. Note that a formula in $L_{\omega_1\omega}$ is a countable sequence. If $\cal M$ is a countable model then it might just happen that it does not contain that particular formula. So you might want to require something else, for example if $\sf ZFC$ proves that there is such formula, in which case different models (which may also have different $\omega_1$) may disagree on the contents of the formula, but will agree on its existence. In either case the answer, I believe, is negative. I'll think about it, though. – Asaf Karagila Dec 15 '13 at 16:39
  • A different question: if we take a formula $\Psi$ in $L_{\omega_1,\omega}$ such that $\Psi$ is a countable conjunction of formulas indexed by natural numbers (and depending only on the indices), does $\Psi$ exist in arbitrary countable models of set theory? – user48900 Dec 15 '13 at 16:43
  • If a model has non-standard integers, then no. Because that formula, being a conjunction of things depend only on the natural numbers used, can be used to decode the set of standard integers. – Asaf Karagila Dec 15 '13 at 16:46
  • If we assume that the model does not have non-standard integer the answer will be yes? – user48900 Dec 15 '13 at 16:47
  • I'm not sure... – Asaf Karagila Dec 15 '13 at 16:49
  • This is a different spin on the problem, but is it enough to show that $\mathbb R$ is the unique Archimedean real closed field? Then if $T=RCF$, then $\bigwedge T \wedge \forall x,y \bigvee_{n<\omega} x<ny$ (where $ny:=y+\ldots+y$ $n$-times) should do the trick. – Maxwell Dec 15 '13 at 16:53

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