Hello Terren,

On 18 Aug 2011, at 03:52, Terren Suydam wrote:

Hey Bruno, Given a machine's inability to prove its own consistency,

OK. That is Gödel's second incompleteness theorem.

and how this result gives rise to the many logical distinctions that map to the hypostases (per Plotinus) as you've written, then I wonder what you would say to this: if a machine is universal, surely it can run a program that implements a (higher order or more-encompassing) logical machine that *can* prove the machine's consistency. If so, can't it use that result to prove its own consistency?

`Good question. The answer is no, of course. The machine would prove`

`its own consistency, and becomes inconsistent (by Gödel's second`

`incompleteness theorem).`

RA(*) cannot prove its own consistency (no consistent machine can do). PA (a much stronger LUM than the UM RA) can prove the consistency of RA.

`RA can implement and run PA. Or, RA can emulate PA. But why would RA`

`believes in PA's axiom and theorem?`

`That would be an error à-la Searle's chinese room. With enough time`

`and patience I can emulate the brain of a chinese guy, but this would`

`not entail that I understand chinese.`

`I can emulate the brain of Einstein, but that does not mean I agree or`

`even understand what Einstein is saying.`

`In math the key here is that computability is absolute and proof is`

`relative. Searle makes a confusion of level.`

`From the fact that PA can emulate ZF doing a proof of the consistency`

`of PA, PA cannot deduce its own consistency, because there is no`

`reason for PA having the slight understanding of ZF argument.`

`All right? This is very important, as you guess. The Universal`

`dovetailer is more or less equivalent, with respect to emulability,`

`to RA, and it emulates all LUMs, but is not a LUM itself.`

Bruno (*) Acronym explained: RA = Robinson Arithmetic PA = Peano Arithmetic ZF = Zermelo-Fraenkel Set Theory.

`UM = Universal machine (like RA, PA and ZF). I identify machine and`

`theory (technically this can be done properly thanks to a theorem due`

`to Craig).`

LUM = Löbian Universal Machine (like PA and ZF) Bruno 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.