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/ 
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.




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