Bruno: Is there a UD that is implemented in Fortran? Ronald

## Advertising

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.