Re: Proportions of Infinity

From: Bruno Marchal <marchal.domain.name.hidden>
Date: Fri, 23 May 2003 10:25:37 +0200

At 11:52 21/05/03 -0700, Hal Finney wrote:
>Maybe you can say that programs could be (countably) infinite in length.
>I'm not sure it makes sense but it's possible.

I am not sure it is interesting but it's possible.



>However, if you adopt this convention, then I think you can show that
>a dovetailer cannot run all programs. It can only run a countable number
>of programs since it can only run a countable number of steps. But
>if programs are infinitely long, then there are an uncountable number
>of programs. There are uncountably infinitely many programs that the
>dovetailer never runs.

It depends what you mean by "running programs". Suppose for simplicity
that each infinite binary sequence is an "infinite program". If you accept that
running a prefix like 10101100001, where each 1 and 0 are seen as instruction,
is a way to run at once all the programs which begins by that prefix, then you
can give a sense to dovetailing execution on uncountable set of infinite
programs.
(It is the same argument I have used to explain in what sense the UD
"generates"
all the reals, where indeed at each steps the UD names uncountably many reals.
And this not only makes sense but plays an unavoidable role when we look
for the
measure on first person (plural) experiences.
Of course generating all the reals in that sense has nothing to do with the
fact that
we cannot put the reals in 1-1 correspondence with the natural numbers.
OK?

Bruno
Received on Fri May 23 2003 - 04:37:05 PDT