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