Bruno:
yes that is unfortunately true.
Ronald
On Dec 30, 10:25 am, Bruno Marchal <marc...@ulb.ac.be> wrote:
> 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/
>
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