On 25 Aug 2012, at 07:30, Stephen P. King wrote:

## Advertising

On 8/24/2012 12:02 PM, Bruno Marchal wrote:As emulator (computing machine) Robinson Arithmetic can simulateexactly Peano Arithmetic, even as a prover. So for example Robinsonarithmetic can prove that Peano arithmetic proves the consistencyof Robinson Arithmetic.But you cannot conclude from that that Robinson Arithmetic canprove its own consistency. That would contradict Gödel II. When PAuses the induction axiom, RA might just say "huh", and apply it forthe sake of the emulation without any inner conviction.With Church thesis computing is an absolute notion, and alluniversal machine computes the same functions, and can compute themin the same manner as all other machines so that the notion ofemulation (of processes) is also absolute.But, proving, believing, knowing, defining, etc. Are not absolute,and are all relative to the system actually doing the proof, or theknowing. Once such notion are, even just approximated semi-axiomatically, they define complex lattices or partial orders ofunequivalent classes of machines, having very often transfiniteorder type, like proving for example, for which there is a branchof mathematical logic, known as Ordinal Analysis, which measuresthe strength of theories by a constructive ordinal. PA's strengthis well now as being the ordinal epsilon zero, that is omega [4]omega (= omega^omega^omega^...) as discovered by Gentzen).Dear Bruno,What happens when we take the notion of a system to those thatare not constructable by finite means? What happens to the proving,believing, knowing defining, interviewing, etc.?

`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.`

`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.`

`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.`

Bruno

-- Onward! Stephen http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html --You received this message because you are subscribed to the GoogleGroups "Everything List" group.To post to this group, send email to everything-list@googlegroups.com.To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com.For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.

http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.