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Harvey Friedman's "Concrete Mathematical Incompleteness" at http://www.math.osu.edu/~friedman.8/pdf/0.Intro061311.pdf cites the Hasse Minkowski theorem saying quadratic forms over a number field are equivalent if and only if they are equivalent over every completion of the field (real, complex, or $p$-adic). He says "It would appear that using standard techniques, this can be put into" first order arithmetic. He asks whether it or some stronger theorem can be made $\Pi^0_2$ or even $\Pi^0_1$.

Is there published work on this problem?

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  • $\begingroup$ The standard methods he mentions are standard in practice. The usual proofs of Hasse-Minkowski over $\mathbb{Q}$ (the case for other number fields is similar) reduce the $p$-adic calculation for any given rational quadratic form to calculations mod $p^n$ for $n$ specifiable from the coefficients of the form. And that is the usual point of using the theorem. But has anyone gone into the specific quantifier complexity of the theorem? $\endgroup$ – Colin McLarty Aug 3 '13 at 1:20
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The community has spoken by silence. No one has worked on this.

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    $\begingroup$ To allude to a standard joke about mathematicians: you know that there is at least one sheep one side of which is black. A deduction about all the sheep in Scotland may be unwarranted. $\endgroup$ – Pete L. Clark Aug 6 '13 at 9:20
  • $\begingroup$ This community is not sheep. $\endgroup$ – Colin McLarty Aug 6 '13 at 22:21

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