Infinite Sets

[danbaron]

I am not a mathematician.

But, anyway, I assert that it is impossible to randomly select an item from an infinite set.

Am I correct?

(If anyone replies, I will explain my reasoning.)

Dan.

[ErosOlmi]

Well, numbers are infinite but on the CPU/memory numbers are stored in a location that, even if it is big, it has a finite physical bounds. So with programming you have always to go down to some compromise.

The following is current thinBasic bounds on numbers: http://www.thinbasic.com/public/products/thinBasic/help/html/numericvariables.htm

Can you give more details? Maybe I'm not understanding well what you need.

Ciao
Eros

[danbaron]

I was just thinking about infinite sets, generally.

For instance, consider the infinite set of non-negative integers, 0,1,2,..

Let's say each succeeding bit in our computer sets the next higher power of 2.
So, the first bit would be for 2^0. The second bit would be for 2^1. The third
bit would be for 2^2, etc.

Say, that we can randomly set each bit. (I think it is possible to randomly (not
pseudo-randomly) set a bit, by using electronic white noise.)

Like you said, in order to randomly select one member from the set of
non-negative integers, we would have to set an infinity of bits. And,
additionally, even if we could set an infinity of bits, I think the integer
selected would always be infinity. Because, if the integer was finite, it would
mean that at some point during the bit setting, all of the remaining infinity of
bits (each higher power of 2), would have to be 0. And the probability of that
happening, is 0.

Think about any continuous random distribution, say, crd(x). Since it is
continuous, it has an infinity of possible values. The probability that a random
variable is in a particular interval (a,b), is the integral of crd(x)dx, from
'a' to 'b'. But, the probability that a random variable has a particular value,
say, 'c', is the integral of crd(x)dx, from 'c' to 'c', which is 0. So, I think
in any infinite set, the probability that some particular value is randomly
selected, is exactly 0.

But, you might say, what if I was suspended over an infinite sea of ping pong
balls, and the balls were numbered, 0,1,2,..(assuming that somehow a number
could be written on a ball, no matter how large the number)? If all of the balls
were mixed up, couldn't I reach down and randomly select one? I think the answer
is, "no, you couldn't". If I am correct, it is impossible to mix an infinity of
balls. No matter how many balls were mixed together, there would still be an
infinity of balls that had not been mixed, i.e., infinity minus any finite
amount, is infinity.

So, I think that, practically (not theoretically), we are, and always will be,
limited to using finite sets.

Dan

[ErosOlmi]

Dan,

I hope someone more clever/prepared than me will help here with your request.

What field/area of competence are you?

Eros

[Charles Pegge]

The only kind of infinity a PC will support is a floating point number divided by 0. Most programming languages will gag when this is encountered at run-time but the Floating point processor will in fact support infinity, both positive and negative.

This is useful when computing atan + or - 90 degrees.

Unfortunately the FPU tan(pi*.5) does not return infinity, only a large negative number which may prove disastrous when attempting to land your plane under PC control. :shock:

Charles

[ErosOlmi]

... which may prove disastrous when attempting to land your plane under PC control. :shock:


:D that image of what can happen with computer is so clearly understandable ...

[LanceGary]

I am not a mathematician.

But, anyway, I assert that it is impossible to randomly select an item from an infinite set.

Am I correct?

(If anyone replies, I will explain my reasoning.)

Dan.



There is an infinite number of real numbers between 0 and 1.

Suppose we define a square area bounded two number lines each running from 0 to 1.

Suppose further we blindfold ourselves and toss a dart in the general direction of the square area until the dart lands. Then as precisely as possible we measure either the X or the Y coordinate of the point where the dart landed.

The number selected would be a random number chosen from an infinite set.

The main limit on this idea would be the precision with which we could measure the exact X or Y coordinate. The atomic theory tells us that (unlike the number line) any physical surface can only be subdivided up to a point. We could of course compensate by making the area bigger and re-tossing the dart, but the area could not be infinitely large. But for all practical purposes, yest we can choose a random number from an infinite set.

Lance

[danbaron]

My training is in continuum mechanics, Eros.

Now, concerning the interval from 0 to 1.

(We'll just consider a physical line segment. The proof is analogous for a physical square area.)

We'll consider the closed interval, [0,1], the inclusive interval, 0 <= x <= 1.

(But, we could just as well consider the open interval, (0,1), the exclusive
interval, 0 < x < 1, it doesn't matter.)

I agree that, theoretically, there are an infinity of real numbers in that
interval.

But, there are qualifications.

If we are only able to measure to one decimal place, then, physically, for us,
there are only 11 real numbers in the interval, 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6,
0.7, 0.8, 0.9, 1.0.

If we are able to measure to three decimal places, then physically, for us,
there are 1001 real numbers in the interval.

If we are able to measure to 10^1000 decimal places, then physically, for us,
there are 10^10^1000 + 1 real numbers in the interval.

In order for there to physically be for us, an infinity of real numbers in the
interval, we would have to be able to measure to an infinity of decimal places.

There are three possibilities for the physical straight line that we measure;
either it is semi-infinite, infinite, or finite.

If the line is semi-infinite, then it only has one end. It begins at some point,
and then from that point, extends forever. In that case, it is impossible to
subdivide the line. So we would be unable to measure the coordinate of a point.

If the line is infinite, then it has no ends. It just extends forever in both
directions. In that case, it is again impossible to subdivide the line. And
again, we would be unable to measure the coordinate of a point.

If the line is finite, then it has two ends. In that case, the line can be
subdivided. We can measure the coordinate of a point. But, we are limited to
dividing the line into a finite number of subdivisions, and therefore, to
measuring the coordinate of a point, to a finite number of decimal places. So
physically (not intellectually), we are constrained to using finite sets. And
therefore, it is impossible to randomly select an item from an infinite set.

Can we show that we are only able to subdivide a finite line segment a finite
number of times? Yes, we can. We can show it by contradiction. We'll assume that
we can subdivide a finite line segment infinitely, and then show that doing
that, is impossible. Let the length of the finite line segment be called, "L".
When the number of subdivisions (and correspondingly, decimal places) is
infinity, what must be the length of a subdivision? It is the limit, as "n" goes
to infinity, of L/n. And for any finite length, "L", that quantity, is exactly,
0. And, it is impossible to have a subdivision with length 0.

Dan