# Re: Simple proof that our intelligence transcends that of computers

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On 25 Aug 2012, at 07:30, Stephen P. King wrote:```
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```On 8/24/2012 12:02 PM, Bruno Marchal wrote:
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As emulator (computing machine) Robinson Arithmetic can simulate exactly Peano Arithmetic, even as a prover. So for example Robinson arithmetic can prove that Peano arithmetic proves the consistency of Robinson Arithmetic. But you cannot conclude from that that Robinson Arithmetic can prove its own consistency. That would contradict Gödel II. When PA uses the induction axiom, RA might just say "huh", and apply it for the sake of the emulation without any inner conviction.
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With Church thesis computing is an absolute notion, and all universal machine computes the same functions, and can compute them in the same manner as all other machines so that the notion of emulation (of processes) is also absolute.
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But, proving, believing, knowing, defining, etc. Are not absolute, and are all relative to the system actually doing the proof, or the knowing. Once such notion are, even just approximated semi- axiomatically, they define complex lattices or partial orders of unequivalent classes of machines, having very often transfinite order type, like proving for example, for which there is a branch of mathematical logic, known as Ordinal Analysis, which measures the strength of theories by a constructive ordinal. PA's strength is well now as being the ordinal epsilon zero, that is omega [4] omega (= omega^omega^omega^...) as discovered by Gentzen).
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```Dear Bruno,

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What happens when we take the notion of a system to those that are not constructable by finite means? What happens to the proving, believing, knowing defining, interviewing, etc.?
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Amazingly, not a lot. That is why I say sometimes that comp can be weakened a lot. G and G* are sound, not only for PA and ZF (which is terribly more powerful than PA, with respect to provability, but, I repeat, the same for computability). If you allow provability to be even more powerful, and accept infinite inference rule, like the omega- rule in analysis, or some axiomatic form of second order logic, or even more non constructive, G and G* will still remains correct and complete.
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If you continue on that path, G and G* will remain correct, but no more complete. That is the case if you define provability by satisfied by some models of a rich theory. By Gödel completeness, satified in all models of the theory, gives the usual provability. But satisfaction by certain models leads to entities needing some supllementary axioms to be added on G and G*. But the present comp theory does not use completeness of G and G*, only the correctness, and so .... you need to go really quite close to God, for avoding the consequences of the arithmetical hypostases.
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Now, to prove this is quite difficult. Solovay announced many of this without proof, and the book by Boolos, the 1993 version gives the detailed proof, but it is technically hard. I use comp, for reason of simplicity, but it can be weakened a lot. I suspect that the real needed axiom is just the assumption of self-duplicability, and not digitalness.
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Bruno

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Onward!

Stephen

http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html

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

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