[chiralSPO (OP)]

There is something that has been bothering me for quite some time now, and I would like to try some discussion of it here.

I have some trepidation starting a thread about infinity, as it is a very difficult topic, but I trust that the discussion will at least be interesting.

There is a commonly presented notion that infinite sets must contain everything. It often comes up when someone claims that infinitely many monkeys each with a typewriter and an infinite amount of time typing will eventually reproduce the entire works of Shakspeare (as well as every version of the bible, including one in which every mention of God is replaced with the word Spatula. etc.) Or it is said that the multiverse theory implies infinitely many different variations of the universe, representing all of the things that are possible. Or that any finite string of digits that one can imagine must be present in the digits of π (or e or φ) etc.

I have always felt that there was some logical fallacy in this line of reasoning, but I cannot quite prove to myself that there is a contradiction.

When trying to think of a counterargument, I usually start with something along the lines of:
I can come up with infinitely many infinitely long strings of numbers that are non-repeating, but never contain a certain string of numbers. For instance 0.1234567890011223344556678899000111222333444555666777888999000... will never ever contain the string 2468 in it.

Now, I think that the problem this argument has is that the example I used is clearly not random. However, I could make adjustments to the algorithm that I use to generate these strings of numbers such that they are effectively random--equal ratios of each digit, and each dyad and each triad (pairs and trios of digits), and no way to predict the next number in a string given all previous digits, other than the number won't contain a single instance of finite string X. The longer finite string X is, the easier it is to generate an algorithm that is otherwise random.

I suppose the counter-argument to that is that "effectively random" and "actually random" are not the same thing. If each digit truly has a random distribution within the string, then any string of finite length must have a finite nonzero probability of happening, and that therefore with infinitely many digits the expected number of times the string appears would have to be infinite (any finite, nonzero number times infinity must be infinite).

Thoughts?

[Bored chemist]

Strictly, the monkeys will write any finite set of words.
It's because, if they haven't written any (finite) test script, it's because you haven't waited long enough.

Also, if they are using a qwerty keyboard, they will never write a copy of the original Bible- because it was written in a different alphabet. That may be a better mimic of the series that doesn't include 2468 because the "rules" of the monkeys exclude it.
The monkeys are meant to represent a "random"  set of keystrokes.

[jeffreyH]

How far apart from each other are the monkeys and typewriters? Is it an infinite distance? Maybe Hilbert's infinite monkey sanctuary.

[evan_au]

Professor Brian Cox's Infinite Monkey Cage has already come up with some variations on Shakespeare and the Bible.

The fact that they have only come up with short texts so far is that they have only generated 99 episodes to date.

See: http://www.bbc.co.uk/programmes/b00snr0w/episodes/downloads

[chiralSPO (OP)]

I guess I just have to accept this unusual consequence of such an unusual scenario. It's straightforward enough to prove that, for any event with a nonzero chance of occurring, the probability of the event actually manifesting tends to 1 as the number of trials tends to infinity. (Though I will note that it is straight-forward to "prove" many things using infinity that are not actually true, and are actually obscuring logical fallacies in the confusion of infinity... so I just want to make sure that this is not one of those instances.)

[jeffreyH]

There is a non zero probability of a monkey pressing the same key an infinite number of times. How meaningful is this?

[chiralSPO (OP)]

There is a non zero probability of a monkey pressing the same key an infinite number of times. How meaningful is this?

Actually, I think there is a probability of zero for any infinite string of keys (only finite strings will appear with a nonzero probability).

This can be shown by considering the limit as n approaches ∞ of the probability of a string being generated (typed). If we limit ourselves to 26 letters of the alphabet, then the probability of repeating the same letter every time is (1/26)^n–1 (allowing the first letter to be any of the 26, but then requiring all subsequent letters to be the same as the first.) As n approaches ∞, (1/26)^n–1 definitely approaches 0. (even by the 10th repitition, we are down to 7x10^–15)

An interesting paradox arises here: If a random string of numbers is generated, and is infinitely long (countably infinite), the probability of any specific infinite string is 0. Every infinitely long string has a 0 probability of being generated randomly. But, if you consider all of the uncountably infinitely many "possible" strings, there is still a probability of 1, that the string generated is contained within that set.

For example, if we consider selecting a single number from a uniform distribution of the continuous number line between 0 and 1 (including all rational and irrational numbers), it's guaranteed that the number selected had a 0% chance of being selected!

I think many of these apparent paradoxes come up because people are playing fast and loose with the concept of infinity, trying to plug it in as a number and do arithmetic, rather than solving for the limits in each case....

[Humanism_at_its_finest]

Some Simple and easy to understand Math:
The number of combinations you can make with 10 characters in 3 digits is= 10x10x10 (10^3) So there are 26 (+1 for a space) letters of the English alphabet so in a 3 letter long term there are= 27x27x27 (27^3) combinations. So, of course, ∞ monkies would produce an infinitely long term of letters which would be= 27^∞ combination. Which of course = infinity. And there is your final result, infinite combinations. The bible is just one combination of letters and if we have infinite combinations at some point (it might take infinite years to happen it will still happen) we will have a bible printed neatly by a group of Bonobos!

[petelamana]

There is a non zero probability of a monkey pressing the same key an infinite number of times.

What if the monkey dies at the typewriter and his finger falls in death on a key?

Pondering the scale and scope of the infinite is tantamount to pondering the existence of the divine.  You have touched upon a key distinction possibly unique to man.  I once read that the moment we believe we have a pure definition of what/who God is, is the moment God changes.  Or, I suppose, to put it in a rudimentary form:  the moment we declare x = y, y changes. 

All we can be certain of is the essentials of mathematics, for as it is quoted in the film Pacific Rim:  "Numbers do not lie. Politics and poetry, promises, these are lies. Numbers are as close as we get to the handwriting of god."