find an error in this mathamatical paradox?

Question

If their was such a think as a random number genorator (RNG) that picked any random number from zero upwards. no limit.

The chance of the RNG picking a previosly selected number is 1/infineity... which effectivly = 0

the chance of it picking any number from 0 to 1,000,000,000,000 = 1,000,000,000,000/infinity = 0

any number divided by infinity is effectivly zero.

but because the RNG can pick a number infinitly large, it could pick infinity

and infitity/infinity = 1

therefore, a RNG that genorates any number larger than zero must genorate and infineitly large number EVERY time.

therefore it is not random at all because it can not pick a finite number.

so its a paradox

Answer 1

infinity/infinity = any number you want (or even numbers you don't want).

The LIMIT (as x grows to infinity) of any finite number divided by x is zero. A number divided by infinity is not defined.

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All the "random" number generators I know are "pseudo" random number generators. They have limits (they are bounded). They use well-defined functions (including, for example, the logarithm of the remainder of the local oscilator's cycle in relation to seconds of time) which they then multiply by some integer picked (pseudo-randomly) from a list or provided by the user or a program.

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You are correct in assuming that the number generator you describe should rarely give an integer we can pronounce...

However, the correct approach is: the probability that the chosen number falls in the interval [a, b) is zero, for any integer a and b, with b greater than a.

Because such a thing is "proven" with statistical analysis, it does not forbid the RNG from generating an integer that you could recognize. All you can prove is that if you let the RNG run long enough, the probability tends to 0%.

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A variant is that if a random process can yield any rational number, then the probability that the chosen number is an integer should tend to zero.

Answer 2

to tell you the truth, I agree with every other answer on this page; however, the actual first place you went wrong is where the RNG picks a number from 0 to 1,000,000,000,000 or 10^12, the chance of picking one number in that domain is 1/((10^12) + 1) NOT 10^12/infinity, and it is strictly such that because one number has to be chosen.

Answer 3

First of all there is a problem in your wording because infinity is a concept not a number. You may have been trying to use contors trasfinite numbers, such as ℵ0, which is infinately large.
Then there are 2 problems. 1/ℵ0 does not equal zero, it is an infinately small number. Also, what you say as infinity/infinity does not neccesarily equal one, because there are different sizes of that which is infinite. for a review of the basic ideas of infinite numbers listen to the first few episodes of the podcast "The Math Factor" which can be found on iTunes.
Hope this helps!

Answer 4

"but because the RNG can pick a number infinitly large, it could pick infinity"

This is wrong, because "infinity" is not a number. There's no largest number that the RNG can pick, but any number that it does pick will be an actual number, not infinity.

Likewise, infinity/infinity doesn't mean anything, except as calculus shorthand. And you can consider taking the limit as x goes to infinite of x/(x squared) to see that that shorthand doesn't always correspond to 1.

Answer 5

There is no paradox - such a random number generator can't exist.

Note: If it's not a uniform distribution, you could have it. For example, if you had a 1/2 chance of returning 0, 1/4 of returning 1, 1/8 of returning 2, 1/16 of returning 3, and so on (1/(2^(n+1)) chance of returning n), this would not be a paradox because the probabilities add to 1.

Answer 6

you are wrong. There is no number called infinity
Your RNG can pick any number [0, infinity), that's it.

By the way infinity/infinity is not 1, it equals anything you want. For example, I want it to be 50. so let
y = 50x
then lim y/x as x ->infinity = infinity/infinity
but y/x = 50x/x = 50
so infinity/infinity = 50, or anything you want.

Answer 7

You have to be careful with your argument because infinity/infinity is not necassarily one. There are infinite levels of infinity

Answer 8

infinity divided is undertermined, not one,

gotta use L'hospitals rule for that