On 28 Dec 2013, at 17:35, Stephen Paul King wrote:

Dear Bruno,

On Sat, Dec 28, 2013 at 7:09 AM, Bruno Marchal <marc...@ulb.ac.be> wrote:

On 28 Dec 2013, at 04:56, Jason Resch wrote:

On Fri, Dec 27, 2013 at 10:42 PM, Stephen Paul King <stephe...@provensecure.com > wrote:
Hi Jason,

"Any program, and whether or not it ever terminates can be translated to a statement concerning numbers in arithmetic. Thus mathematical truth captures the facts concerning whether or not any program executes forever, and what all of its intermediate states are. "

this also captures every instance of random numbers as well.

It is not clear to me what "random" means in arithmetical truth.

Randomness can appear from the perspectives of observers, but I don't see how it can arise in arithmetic.


It appears in all numbers written in any base. Most numbers are already random (even incompressible). I guess you know that. In the phi_i(j) in the UD, randomness can appear in the many j used as input, as we usually dovetail on the function of one variable. (but such input can easily be internalized in 0-variable programs).

OK, I must agree, but can you see how this removes our ability to use the natural ordering of the integers as an explanation of the appearance of time?

Of the physical time? yes, that is right. That is a consequence of the delay invariance of the FPI. But we can still use it indirectly. It is part of the additive-multiplicative structure of the numbers that we assume (through the numbers laws).

Since there are multiple and equivalent (as to their properties) sequences of integers that have very different orders relative to each other, if we use these ordering as our "time" we would have a different dimension of time for every one!

On the contrary. As you just said, the appearance of time is not dependent on that order.

For a long time I got opponent saying that we cannot generate computationally a random number, and that is right, if we want generate only that numbers. but a simple counting algorithm generating all numbers, 0, 1, 2, .... 6999500235148668, ... generates all random finite incompressible strings, and even all the infinite one (for the 1p view, notably).

In that (trivial) sense, arithmetic contains a lot of 3p randomness, even perhaps too much. Then 1p randomeness appears too, by the 1p indeterminacy (and that one is in the eyes of the machine).

Chaitin's results can also explain why we cannot filter out that 3p randomness from arithmetic.

Have you had any more thoughts on the "book keeping" problem we have discussed in the past?

Can you remind me? Thanks.



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