Axiomatizable Theory: Proving T is Axiomatizable

Dragonfall · 2008-04-28T04:04:33+00:00

Homework Statement Suppose T is recursively enumerable. Show that T is actually axiomatizable (there is a recursive set P such that Theory of P = T). The Attempt at a Solution The converse of the question is easy to show, by using compactness. But I have no idea how to do this one.

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...

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Homework Statement

The Attempt at a Solution

Thread 'Prove that the integral is equal to ##\pi^2/8##'

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