On 02 Dec 2011, at 13:26, Stephen P. King wrote:

Dear Bruno,Please re-read that last post slowly. I fear that you arerushing to a judgement of what I am trying to communicate andcompletely missing the idea that I am trying to sketch out.On 12/2/2011 6:26 AM, Bruno Marchal wrote:On 02 Dec 2011, at 05:16, Stephen P. King wrote:snipWorse, there will be no effective means (and thus no completetheory of universal machine or language) to decide if some f_i isdefined on N or a proper subset of N. If that was the case, wewould be able to filter out the functions from a proper subset ofN to N from the functions from N to N, and then thediagonalization above would lead to 0 = 1.Let us call a function partial if that function if either from Nto N, or from a subset of N to N, and a function is total if itis defined on N. The reasoning above shows that the set of totalfunctions is not immune to diagonalization. But the superset ofthe partial functions is, and that is a deep strong argument forChurch thesis.But it still seems that there is something missing in thisconclusion about the Church thesis. It is that for computation allthat one needs to consider in a model is the N -> N and n /subsetN -> N maps/functions. The problem of time that I have complainedabout is part of this problem that I see.Time is not relevant here. We need only the natural numbers.Your notion of time seems to be lifted directly from theordering of the natural numbers, so in a sense you are taking timeas the sequence 1 2 3 4 .. as inducing a ab initio measure ofchange. The problem is that unmeasurable sets exists and restrictingourselves to tiny little islands of thought is not progressive. Wemay find solutions to the measure problem by considering morecarefully exactly how we define measures. Ordinary notions ofmeasure seem to be based on Platonic notions, definitions given byfiat. Are you familiar with how topos theory treats the notion of aset? The following is from Jonhstone's Topos theory" p. xvii<cfbedije.png> I am considering an evolving universe, not a "fixed" one.

`The notion of classical computability is topos independent. Non`

`classical computability is interesting but not relevant for the issue`

`of the classical Church thesis that we need to make sense of comp in`

`cognitive science. The notion of time is also not relevant.`

`The role of topos is comp is explained in "conscience et mécanisme". A`

`topos basically is good to modelize a mathematician's mind, not the`

`arithmetical reality.`

It reminds me strongly of the problem of the axiom of choice,The axiom of choice is not relevant here. This can be made precise:ZF and ZFC proves the same arithmetical truth. I don't use settheory at all.Are not the natural numbers a set and thus have an implicit settheory?

`Natural numbers are no more set than fortran program. You can`

`represent numbers with set, but this does not make a number a set.`

`Natural numbers admit a simpler first order theory than set or`

`elementary toposes.`

Just because you did not invoke a set theory specifically does notabsolve you from the implicit use of a set theory.

`I don't, in the theory. I do in the epistemology at the naive level,`

`like engineers, philosophers or like when I do shopping. Set theory is`

`just not relevant for Church thesis.`

I write "a set theory" instead of "set theory" because set theoriesare Legion! I have seen instances where huge fights have occurred inacademia because of people having completely different set theorieswith which they are interpreting a theory.

`That is one good reason to not use set theory. I don't believe in`

`sets, actually. But even this remark is not relevant for what we were`

`talking about.`

where the existence of unmeasurable sets cannot be excludedinducing such things as the Banach-Tarski paradox. The sameproblem that occurs in the result you discuss above: there is afunction/object that cannot be exactly defined.Which one? You confuse computation and definition.When are definitions computable (constructable by recursivefunctions) and when are they otherwise? This is not a trivial point!

That is a reason for keeping those notion apart.

How do we get around this impasse?I don't see an impasse. I explain old and basic uncontroversialnotions.LOL! It seems that I am more radical in my thinking and you areconservative in yours here.

`I have never hide that I am an extreme conservative. Modernity has`

`ended in 523.`

The problem is that these "old and basic uncontroversial notions"are causing problems for the advancement of physics and ourunderstanding of "old" problems, such as the mind-body problem.

`Indeed, I show that those old and uncontroversial notions makes`

`physics a branch of number theory. Just study and criticize the proof,`

`instead of escaping forward with technical notions none knows in the`

`list. Or use them specifically.`

What if there is a way to sequester the pathological parts of thefunction without having to define the function?This is too vague.I agree, my apologies, but if I knew the explicit well formedstatement required then I would not be asking the question that I amasking.(Something like this is discussed here http://ndpr.nd.edu/news/24157-on-preserving-essays-on-preservationism-and-paraconsistent-logic/)Nothing there is relevant with the issue I was talking about.It is relevant to the question I am asking. Please re-read myoriginal post and try to see the idea that I am considering.

I did. I don't see the point at all. Sorry.

We see a similar process in physics where the abstract spacesare defined to be well behaved ab initio. What if this "goodbehavior" is emergent, like what happens when we couple a largenumber of chaotic systems to each other. The entrainment processbetween them forces them to behave as if they are nice and linear!(L. M. Pecora and T. L. Carroll, “Synchronization in chaoticsystems,” Phys. Rev. Lett. 64, 821 (1990). )?Are you familiar with configuration, state and phase spaces usedin physics?

`What is the point of this with Church's thesis? Church's thesis has`

`nothing to do with physics. Don't confuse Church's thesis and Deutsch`

`physicalist version (which is a different thing).`

As far as I know, in mathematics, only computability, andcomputability related notion (like their relativization onoracles) are immune for the diagonalization. Gödel, in hisPrinceton lecture called that immunity a miracle.Immune from diagonalization but not from forcing.? (forcing is a technic in set theory or in model theory. It doesnot concern us here). Modal logic, which have a role incomputability (but not here) generalizes forcing in some way.Do you see that I am considering generalizing the notion ofcomputation and with it the notion of information? The currentnotion of information (Shannon) is based on the differences thatmake a difference for physical systems and thus assumes a particularform of physics. This is what D. Deutsch mentions in his writingsthat you seem to argue is wrong, but I think it is you that arethinking too narrowly about what is physics.

`Physics is not part of the theory. I presume nothing. And the result`

`is that physics is a number dream. But that is a result in a theory.`

It seems to me, and I admit that this is just an intuition, thatwe are constraining the notion of computation to too small of abox to realize its full potential. Why is the notion of time soscary for computer scientists and logicians?It is not scary, and it plays some role in some chapter. but wegeneralize it by using any recursive function on the inputs andprograms. It is hard to make sense of your comment. You introducedifficulties where they don't exist, I think.No, I see short-comings in the current set of conceptual toolsthat we use to build explanations. I am trying to explore furtherout in the undiscovered country...Everything that I explain depends on this. UDA works thanks tothe universality of the UD, which relies on that diagonalizationclosure, and AUDA uses the fact that self-reference is build onthe diagonalization procedure. Unfortunately, or fortunately,all diagonalizations are hidden in Solovay's theorem on thearithmetical completeness of G and G*.OK, but I want to go further. We need for logics theequivalent of a principle of variation so that we do not have toresort to the ansatz of Platonic walls to explain where universalcomputation is implemented.?You seem to know a something about this idea given your previousreference to amenable groups ,OK, but that has nothing to do with Church thesis.Can you think of the trail of ideas connecting these twoconcepts? You have cut my post into little pieces here and thus haveambiguated the context, please re-read the original post.

`You don't make specific point. I sell you an aquarium, and you`

`complain we cannot put birds in it.`

but it is as if you are afraid to look over the edge and stareinto the abyss. I think that the Tennenbaum theorem offers us aclue. It states that "no countable nonstandard model of Peanoarithmetic (PA) can be recursive".I think that you mix many things. We were not in Model theory.Could we be in a frame that includes both Model theory andcomputer theory and number theory?

Not simultaneously, and without purpose.

We need to retain the property of countability and recursivity forcomputation but need the spectrum of possibilities that non-standard models allows to act as a range of variation. Is there aversion or model of arithmetic that would allow this but retainthe expressiveness of PA?PA. The non standard models are models of PA. I have no clue whatyou are trying to say. You should not mention more technical stuffwhen simpler one are presented to you, unless you can make aspecific point.BrunoIs there something like PA but more expressive that allows for anotion of countability (maps to some set of number) and recursive(so that we can have a more general notion of "countable" and"recursive" in the sense of the quote from Johnstone about sets)?

Yes we can. The question is "why do you want to do that?". 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.