> On 9 Sep 2018, at 01:12, John Clark <[email protected]> wrote: > > On Sun, Sep 2, 2018 at 2:19 PM Bruno Marchal <[email protected] > <mailto:[email protected]>> wrote: > > > A function is computable if we can explain to a dumb (but docile) human > > being how to compute it, on any of its argument. > > And some functions (the Sine function for example)
We talk only about functions from N to N. The computable real functions requires a good understanding first of the computable functions from subset of N to N? > can be proven to be computable and some functions (the Busy Beaver function > for example) can be proven to be non-computable) but there is no general way > to know if any given function is computable or not. >From its code, indeed. That is Rice theorem, and I have just proven it in this >thread. > > > Each f_n is computable (by definition!) and so each f_n(n) can be computed, > > and adding one is certainly computable, so, the only thing which can be NOT > > computable is the bijection itself. This means that the f_i, although > > enumerable, are not recursively enumerable. > If a universal language exist, then it cannot compute the enumeration of all > computable function from N to N. > > The Ackermann function is not primitive recursive and yet it is computable, > like the Busy Beaver the numbers soon become huge (although finite) but > unlike the Busy Beaver a Turing Machine can always calculate them. No problem with this. I avoid using the primitive recursive functions. > > > Then the Löbian machine are those universal machine which knows that they > > are universal, and so get acquainted with the consequences (the infinite, > > the non provable, the non observable, …). > > Nobody on this planet uses the term "Löbian machine" except you. It is just a more precise version of what popular books described by “sufficiently rich theory”. There are many definition, but they are all equivalent. Any Turing complete theory of any universal machine, with sufficiently strong induction axiom (like sigma_1 induction) constitute a Löbian machine. Distinguished feature; their provability predicate verifies Löb’s formula: []([]p->p)->[]p. Easy exercice, show that Löbian machine obeys to Gödel’s second Gödel’s theorem <>t -> ~[]<>t. > Turing explained exactly precisely how to build one of his machines but you > have never given the slightest hint of how to build a "Löbian machine" or > even clearly explained what it can compute that a Turing Machine can’t. ? That means just that you need to go being step 3 in my thesis, or, if you don’t want to do philosophy of mind, just read the mathematical part. The notion of Löbian machine is easy to construct, and the mathematical reality is full of example of Löbian machine, and Löbian god (mathematical object which are not Turing emilable but having still a notion of belief associated with them which still obeys the Löb’s formula. > Can it tell if any given function is computable? Can it find the 8000th Busy > Beaver number? Can it even find the 5th? A Lpobian machine is just a universal machine capable of proving its own universality. Why do you want it to be able to do what a god can do? I don’t like such sentence as it assumes that I would have something like that, but that is simply not the case. I study Mechanism. No machine can ever decide if any code is a code of a total computable function, nor compute a non computable function. > > >Here there is a second miracle, which is that the propositional part of the > >theology is decidable! > > Homemade gibberish. ? > How would things be different if "the propositional part of the theology" > were not decidable? Solovay theorem would be false, and the subject of machine theology would be far more complex. Note that the theology of machine has highly undecidable at the first order level. G and G* are decidable, but qG (provable machine’ logic of belief/proof with quantifier) is PI_2 complete (and thus highly undecidable), and qG* (true machine’s logic of belief/proof* with quantifier) is PI_1 complete in the oracle of truth. Bruno > > John K Clark > > John K Clark > > > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected] > <mailto:[email protected]>. > To post to this group, send email to [email protected] > <mailto:[email protected]>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <https://groups.google.com/d/optout>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
