On 30 Dec 2009, at 03:29, ronaldheld wrote:
> 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
Good exercise. A bit tedious though.
> 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
>>>> of addition, and multiplication. Then the existence of the
>>>> 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
>> (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
>> 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
>> the same sense that prime numbers exists.
>> "exist" is used in sense of first order logic. This leads to the
>> philosophical problems in math, no new one, and the UDA reasoning
>> not depend on the alternative way to solve those philsophical
>> 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
>>>> 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
>>> 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
>> an infinite object, and such object does not exist in simple
>> arithmetical theories. But all finite parts of the UD* exist, and
>> will be enough for "first person" being able to glue the
>> For example, you could, for theoretical purpose, represent all the
>> running of the UD by a specific total computable function. For
>> 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
>>> 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
>>>> Time is not difficult. It is right in the successor axioms of
>>> 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
>> 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
>> 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/
>> 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
>> 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.
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