**From:** J. Andrew Rogers (*andrew@ceruleansystems.com*)

**Date:** Mon Aug 23 2004 - 17:11:59 MDT

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<meta>

I've been flying around the country and therefore mostly

offline for the last few weeks. I'll be leaving again in a

few days, so if I'm slow to respond to things, that is why.

</meta>

As has probably been fairly apparent, I discarded the

notion of infinities having any practical real existence

years ago for a variety of reasons. Therefore, I suppose

I could be called an "infinite set atheist" even though I

wouldn't generally describe it that way. Infinities are

arguably interesting and useful in mathematics, but do not

have a practical application in real systems other than as

crude approximations. Since I concern myself solely with

systems that exist in our universe, I do not have much use

for the notion. This needs to be explained though, because

the above is different than asserting that the universe is

finite state.

The kinds of systems I assume have two basic properties:

1.) They are countable in all dimensions (i.e. discrete).

For our universe, the Planck dimensions work just fine as a

physical analogue of this.

2.) In finite time, the system can only express finite

algorithms. The speed of light serves this function given

#1. (For all I know, this may follow from #1; I have not

thought about it very hard.)

The implied asterisk above is that countable doesn't mandate

finite-ness, and having a system bound to finite-state-ness

in finite time is a perfectly adequate description of real

systems. No point in overly restricting the model when not

necessary, and it doesn't affect the mathematical treatment

of a system. So in a sense, I don't have a problem with

infinities as long as they are never expressible as a

practical matter. In other words, they can be treated as

finite state systems for many purposes even if they

technically are not under the usual mathematical

definitions. You can only have infinite algorithms given

infinite time (and assuming infinite space).

A fun theoretical part is that the limit of K-complexity for

a given computer varies as a function of time, among other

things, a more accurate model generally even for normal

silicon, though it is approximated to a static value. I

cannot recall ever seeing this treated as variable in

theory in this way. Pervasively treating a variable as static in theory

is not unusual, e.g. a similar type of situation exists in

Black-Scholes, as long as one is aware of this approximation so that it

can be discarded when necessary.

This is closely related to the memory latency problem of

hardware. No matter how much RAM you theoretically have,

the size of an algorithm is constrained by the amount of

space you can reference in a finite period time.

Not too much meat here -- I'm on a time restriction -- but

there you have it. I've taken to calling systems that

meet this description as "algorithmically finite", as

I consider them a useful and distinct class of system, but

as far as I've been able to tell there is no term in

mathematics for this particular type of tidy theoretical

system.

j. andrew rogers

**Next message:**Tomaz Kristan: "Re: The cult of infinity"**Previous message:**Tyler Emerson: "Singularity Institute's The SIAI Voice - August 2004"**Next in thread:**Tomaz Kristan: "Re: The cult of infinity"**Reply:**Tomaz Kristan: "Re: The cult of infinity"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ] [ attachment ]

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