# Re: How PIP solves the hard problem of consciousness

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On 26 Sep 2013, at 19:41, meekerdb wrote:```
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```On 9/26/2013 2:12 AM, Bruno Marchal wrote:
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which for a logician are more demanding than the reals, as the first order theory of the real is NOT Turing complete
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Bruno, could point to a pedagogical reference on that?
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i deduce it from a Theorem by Tarski which shows that the first order theory of addition and multiplication on the real is decidable. Any polynomial equation, for example, can be solved by Sturm-Liouville method, when Diophantine polynomial can be Turing universal. Think about x^n+y^n = z^n. There are trivially solutions for all n, and the theory is not complex. The same equation on the natural numbers has many non trivial solutions for n = 2 (already known 6000 BC, apparently), and it took 300 years, and quite advanced mathematics, to show that there in no such non trivial solutions for n bigger than 2.
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Note that if you add a trigonometrical real function, you get turing universality, as you can define the natural numbers by pi-scaling zero of sinus, for example. Waves is how real numbers invoke the natural numbers!
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I am not sure about the reals + addition + multiplication + exponentiation, but I bet it is not yet Turing emulable. But on the complex numbers, reals + addition + multiplication + exponentiation, you get back Turing universality, given that complex exponential can simulate trigonometrical functions.
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I will try to look for some reference, but beside Tarski's paper, and technical textbook I don't see simple pedagogical references.
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Bruno

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thnx, Brent

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http://iridia.ulb.ac.be/~marchal/

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