On 30 Dec 2009, at 03:29, ronaldheld wrote: > Bruno: > Is there a UD that is implemented in Fortran?
I don't know. If you know Fortran, it should be a relatively easy task to implement one. Note that you have still the choice between a fortran program dovetailing on all computations by combinators, or on all computations by LISP programs, or on all proofs of Sigma_1 complete arithmetical sentences, or on all running of game of life patterns, etc. Of you can write a Fortran program executing all Fortran programs. All this will be equivalent. All UD executes all UDs, and this an infinity of times. Good exercise. A bit tedious though. Bruno > > > On Dec 29, 4:55 am, Bruno Marchal <marc...@ulb.ac.be> wrote: >> On 28 Dec 2009, at 21:24, Nick Prince wrote: >> >> >> >>>> Well, it is better to assume just the axiom of, say, Robinson >>>> arithmetic. You assume 0, the successors, s(0), s(s(0)), etc. >>>> You assume some laws, like s(x) = s(y) -> x = y, 0 ≠ s(x), the >>>> laws >>>> of addition, and multiplication. Then the existence of the >>>> universal >>>> machine and the UD follows as consequences. >> >>> Ok so the UD exists (platonically?) >> >> Yes. The UD exists, and its existence can be proved in or by very >> weak >> (not yet Löbian) arithmetical theories, like Robinson Arithmetic. >> The UD exists like the number 733 exists. The proof of its existence >> is even constructive, so it exists even for an intuitionist (non >> platonist). No need of the excluded middle principle. >> >> >> >>>> Better not to conceive them as living in some place. "where" and >>>> "when" are not arithmetical predicate. The UD exists like PI or the >>>> square root of 2. >>>> (Assuming CT of course, to pretend the "U" in the UD is really >>>> universal, with respect to computability). >> >>> Fine so the UD has an objective existence in spite of whatever else >>> exists. >> >> It exists in the sense that we can prove it to exist once we accept >> the statement that 0 is different from all successor (0 ≠ s(x) for >> all x), etc. >> If you accept high school elementary arithmetic, then the UD exists >> in >> the same sense that prime numbers exists. >> "exist" is used in sense of first order logic. This leads to the >> usual >> philosophical problems in math, no new one, and the UDA reasoning >> does >> not depend on the alternative way to solve those philsophical >> problem, >> unless you propose a ultra-finitist solution (which I exclude in comp >> by arithmetical realism). >> >> >> >> >> >> >> >>>> There is a "time order". The most basic one, after the successor >>>> law, >> >>>> is the computational steps of a Universal Dovetailer. >>>> Then you have a (different) time order for each individual >>>> computations generated by the UD, like >> >>>> phi_24 (7)^1, phi_24 (7)^2, phi_24 (7)^3, phi_24 (7)^4, ... >>>> where "phi_i (j)^s" denotes the sth steps of the computation (by >>>> the UD) of the ith programs on input j. >> >>> If the UD was a concrete one like you ran then it would start to >>> generate all programs and execute them all by one step etc. But are >>> you saying that because the UD exists platonically all these >>> programs >>> and each of their steps exist also and hence, by the existence of a >>> successor law they have an implicit time order? >> >> Yes. The UD exist, and is even representable by a number. UD*, the >> complete running of the UD does not exist in that sense, because it >> is >> an infinite object, and such object does not exist in simple >> arithmetical theories. But all finite parts of the UD* exist, and >> this >> will be enough for "first person" being able to glue the >> computations. >> For example, you could, for theoretical purpose, represent all the >> running of the UD by a specific total computable function. For >> example >> by the function F which on n gives the (number representing the) nth >> first steps of the UD*. Then you can use the theorem which asserts >> that all total computable functions are representable in Robinson >> Arithmetic (a tiny fragment of Pean Arithmetic). That theorems is >> proved in detail, for Robinson-ile arithmetic, in Boolos and Jeffrey, >> or in Epstein and Carnielli. In Mendelson book it is done directly in >> Peano Arithmetic. >> >> >> >>>> Then there will be the time generated by first person learning and >>>> which relies eventually on a statistical view on infinities of >>>> computations. >> >>> Is this because we are essentially constructs within these steps? >> >> It is because our "3-we", our bodies, or our bodies descriptions, are >> constructed within these steps. But our first person are not, and no >> finite pieces of the UD can give the "real experience". This is a >> consequence of the first six steps: our next personal experience is >> determined by the whole actual infinity of all the infinitely many >> computations arrive at our current state. (+ step 8, where we abandon >> explicitly the physical supervenience thesis for the computational >> one). >> >> >> >>>> Time is not difficult. It is right in the successor axioms of >>>> arithmetic. >> >>> Here again you confirm the invocation of the successor axioms. >> >> Yes. It is fundamental. I cannot extract those from logic alone. No >> more than I can define addition or multiplication without using the >> successor terms s(-) : >> >> for all x x + 0 = x >> for all x and y x + s(y) = s(x + y) >> >> You have to understand that all the talk on the phi_i and w_i, >> including the existence of universal number >> (EuAxAy phi_u(<x,y>) = phi_x(y)) can be translated in pure first >> order >> arithmetic, using only s, + and *. >> >> I could add some nuances. "To be prime" is an intrinsic property of a >> number. To be a universal number is not intrinsic. To define a >> universal number I have to "arithmetize" the theory. The theory uses >> variables x, y, z, ..., so I will have to represent "to be a >> variable" >> in the theory. The theory "understands" only numbers. I can decide to >> represent the variables by even numbers (for example). "Even(x)" can >> be represented by "Ey(x = s(s(0)) * y)". So "variable(x)" will be >> represented by the same expression. Then I will represent "to be a >> formula", "to be an axiom", to be a proof", "to be a computation", >> using Gödel's arithmetization technic (which is just a form of >> programming in arithmetic). This will lead to a representation of >> being a universal number. >> Now, would I decide to represent the variable in some other way (by >> the odd numbers, for example), the preceding universal number will >> still be in a universal number (intrinsically), but I will not been >> able to see it, or to mention it explicitly. But here, you have to >> just realize (cf the first six step of uda) that the first person >> experience depends on all universal numbers, in all possible sense/ >> arithmetical-implementations. >> In particular "you here and now" are indeed implemented in arithmetic >> in bot the universal numbers based on (variable(x) = even(x), and >> variable(x) = odd(x)). *ALL* universal numbers will compete below >> your >> substitution level. >> >> The fact that elementary (Robinson) arithmetic is already (Turing) >> universal is an impressive not obvious fact. But it is no more >> astonishing than the fact that Conway Game of Life is already Turing >> universal, or that the combinators S and K are Turing universal, etc. >> >> 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-l...@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-l...@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.
