Does anyone know the name/author of the model theory paper where it's proven that you can define the $ < $ relation on $ (\mathbb{Q}, +, \cdot, 1, 0) $?
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Every natural number is the sum of four squares, and so you can define the positive rational numbers as those of the form $(a^2+b^2+c^2+d^2)/(e^2+f^2+g^2+h^2)$, where the denominator is not zero, and this is expressible in your language. And the order is defined by $x\lt y\iff y-x$ is positive.
answered Jan 31, 2011 at 3:50
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$\begingroup$ Yes, I would guess that Tarski knew it. Isn't it used importantly in Robinson's work? $\endgroup$ Commented Jan 31, 2011 at 3:54
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$\begingroup$ Could it be due to (or at least published by) Raphael Robinson? Gerhard "Ask Me About System Design" Paseman, 2011.01.30 $\endgroup$ Commented Jan 31, 2011 at 5:57
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5$\begingroup$ @Joel: A bit easier: Since $p/q=pq/q^2$, it is enough to say that $y-x\ne 0$ and it is sum of four squares. This trick of using Legendre's theorem is by now standard. It is explicitly mentioned by Julia Robinson, in "Definability and Decision Problems in Arithmetic", The Journal of Symbolic Logic, Vol. 14, No. 2, (Jun., 1949), pp. 98-114, in page 109: It is used to show how to define the positive integers inside ${\mathbb Q}$. The definition uses Legendre's theorem, and the fact that the integers are definable, which is her main result. Don't know if it appears in earlier papers. $\endgroup$ Commented Jan 31, 2011 at 6:19
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1$\begingroup$ Andres, thank for the reference, and the simplication! (But I think you mean Lagrange.) $\endgroup$ Commented Jan 31, 2011 at 12:09
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$\begingroup$ Legendre's name is however attached to the much harder result that a positive integer is the sum of three squares iff it is not of the form $4^m(8n + 7)$, and this and similar results in the theory of ternary quadratic forms were used by Robinson in her result on definability of the integers in the field Q. (The streamlined presentation in her thesis doesn't name this result particularly, but see the remarks on page 464 of her Collected Works, particularly about her first breakthrough in this problem. Dan Flath and Stan Wagon had a nice article in the Monthly some years back on this.) $\endgroup$ Commented Jan 31, 2011 at 14:22