Is there a UD that is implemented in Fortran?

On Dec 29, 4:55 am, Bruno Marchal <> 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


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