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.
