Bruno and Brent: "machines" either 'real' numbers' or not, they are humanly devised, even if we state not to be able to 'understand' them. I want to venture into domains where our 'human ways cannot apply e.g. (silly even to attempt to give an examples on whatever we are not capable to knowing) if all the 'topics' we think in, are abstractions from the inter-flowing totality in our attempt to make 'thinkable' models for ourselves. In such case whatever (!) we speak about is "unreal".
Is CT (Goedel?) etc. applicable to the interlacing unlimited net of networks without distinguishable individual topics in a complexity of the totality? How about solving an equation with unlimited variables? Is it 'real' or unreal? Can we - oops: the machine(s) - compute 'infinitiy'? Are 'numbers' limited, or unlimited? In that case how much is infinity minus 1? (and please, do not quote Cantor - he felt free to apply any number of qualitatively different infinities - I seek the limit to 'real'). Another silly question (apologies to Bruno, if it is my forgetfulness): how can we get into the "between" II to III if we have only "I" to work with? (Or between IIIIIIIII(9) to IIIIIIIIII(10) etc.).? Is the number-based world discontinuous? In what way? OUR human way? I like the image of a limitless, continuous complexity of everything in dynamic inter influencing (meaning: un-ceased change) and this does not 'like' discontinua. (Still thinking about understanding Bruno's reply to my recent post ) John On Fri, Mar 18, 2011 at 2:34 PM, meekerdb <[email protected]> wrote: > On 3/18/2011 8:12 AM, Bruno Marchal wrote: > > Hi John, > > In computer science there is something interesting which can be seen as a > critics or as a vindication of what you are saying. That thing is the Church > thesis, also called Church-Turing thesis, (CT) and which has been proposed > independently by Babbage (I have evidence for that), Emil Post (the first if > we forget Babbage), Kleene, Turing, Markov, but not by Church (actually). > > The thesis has many versions. One version is that ALL computable functions > can be defined in term of lambda expressions, or in term of Turing machines, > or in term of Markov algorithm, or in term of Post production system, etc. > All those versions are provably equivalent. > > Such a thesis **seems** to be in opposition with your idea that complete > knowledge is impossible. But it is not. > > The contrary happens. Indeed the thesis concerns only completeness with > respect to computability, and then, as I have already explain on this list, > it entails the incompleteness of any effective knowability concerning just > the world of what machines can do. > > > By "machines" here I assume you mean digital machines/computers. I think > this doesn't apply to machines described by real numbers. But of course we > think it is unlikely that real number machines exist and that the reals are > just a convenient fiction for dealing with arbitrarily fine divisions of > rationals. But there can be a cardinality between the integers and the > reals. I wonder what this implies about computability? > > Brent > > > > Church thesis makes it impossible to find **any** complete theory about > the behavior of machines. I explain this in the first footnote of the > Plotinus' paper. I can explain if someone ask more. It is proved by a > typical use of the (Cantor) diagonalization procedure. > It vindicates what you say, really. We can sum up this by > > Completeness with respect of computability provably entails a strong form > of incompleteness for our means of knowability and provability about > machines' possible behavior. > > This can be proved rigorously in few lines. It is stronger and easier than > Gödel's incompleteness, and it entails Gödel's incompleteness once we can > show that the propositions on the computable function can be translated into > arithmetical propositions (the lengthy tedious part of Gödel's proof). > > Not only Church thesis makes it possible to think about 'everything', but > it makes us able to prove our (machine's) limitation about the knowledge > about that everything. In any case, this makes us modest, because either CT > is wrong and we are incomplete for computability, or CT is true and we are > incomplete about our knowledge about computability, machines, and numbers. > > Best, > > Bruno > > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To post to this group, send email to [email protected]. > To unsubscribe from this group, send email to > [email protected]. > For more options, visit this group at > http://groups.google.com/group/everything-list?hl=en. > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [email protected]. To unsubscribe from this group, send email to [email protected]. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
