Does chess yield the largest meaningful finite number?

This would be true, except for the fact that there are different lengths of infinity. For example, f(x)=2^x > f(y)=2y/10^800, x=y,  for all positive values of X. This means that, for f(∞), although both f(x) and f(y) approach infinity, f(x) is larger. This is the underlying reason for why calculus works, if it were otherwise, x lim x →∞, (2x^2+1)/(x^2+1) would be undefined, not 2.

Sorry for the long, complicated explanation.

Your argument (in paragraph two in the quote) is incorrect, because the same argument would show that there are more rational numbers than natural numbers, because between any two rational numbers, there are infinitely many more rational numbers. However, there aren't more rational numbers than natural numbers, so the argument is incorrect.

The way to think about this in common-sense terms is to realize that when talking of infinities, the normal method of counting (which works well for finite numbers) does not make sense. The method that is used is instead is (conceptually) to try to pair the numbers up in some way. If there's a way to pair them all up with none left over, then they have the same cardinality. See this image for one way that the natural numbers can be paired up with the rationals. There is no way to do this between the natural numbers and the real numbers.

There are more rational numbers than natural numbers, see above.

Both have cardinality ℵ_{0 (\aleph_0).}

Nope, like I said, there are different lengths of infinity, the rational numbers are much bigger, even though they both are infinite.

What is a "length of infinity", exactly?

Edit: Actually, it doesn't matter. Feel free to believe that the cardinality of the rationals is greater than that of the integers or naturals.

Because infinity, by definition, never ends, if one function is increasing at a greater rate than another function, it will be a "longer" infinity, meaning that, although they both go on forever, one goes on for longer. (It's confusing). Like I said, calculus ceters around this.

What makes a number meaningful?

Its social status.

There is a difference between the number of elements in a set and the values in a set. The values in your first function are bigger then the values in your second function. The number of possible values for each function are equal.

Aleph_0(ℵ0) is about the number of possible values for each function, not about their respective values.

The point I was trying to show is how infinity is unreachable. Infinity is imo a dynamic characteristic, which is defined as the unreachable and uncalculable end of the realm of numbers. It is by definition beyond grasp.

In my paradoxical example you can see that the number of elements of N is for a moment bigger than the number of elements of R (the number to express the number of elements in R is bigger than the biggest number in R obviously). That last feature is not true for the Natural numbers: the biggest natural number is a perfect expression of the number of natural numbers. It is the only definition of a set of numbers by which the size of the set is described at the same time. All other types of sets need the set of Natural numbers.

You can prove in a lot of different ways, like the example of Sapientdust, that every other type of set has more elements in it than the set of Natural numbers. To express how much bigger it is, you will need an element of the set of Natural numbers. That specific element is at the time of expression bigger than the size of the set of Natural numbers itself and bigger than the size of the set of elements of the other type. At the moment you can express the difference in size of the set of Natural numbers and the size of for instance the set of Rational numbers, do you need a natural number that is bigger then the biggest number in both sets available. As long as we can think about bigger values are we no way near infinity. We are still on that train, travelling with an ever accelerating speed on its way through finity, but we will never reach infinity. That is not the way to come there.

ℵ0 Is not a specific number, but the smallest beyond-imagination number (thanks FirebrandX) thinkable. That is why I think infinity is a dynamic concept, beyond capability of expression with our finite human minds. Mindboggling beautiful.

Now back on topic: is ℵ0, which is defined as the smallest infinite number, the refutation of the question of the OP or not?

Wouldn't it take less time to say "ohai guys I cant plai chess" instead of posting this retarded game?

@Gil-Gandel: what if you see it as an implicit question that the vast number of chess positions is build upon a huge pile of unmeaningful games?

I thought it was quite humorous.

The possible number of chess moves now has some competition--our Federal Debt...

Go has way more possibilities than Chess. And by way more, I mean in that in astronomical esk units.

You're right, of course. It's still the only explanation I've ever been able to use to get anyone who's not a mathematician or physicist to understand the differences between infinities.

I don't get it.

And that picture with little parenthetical fractions with arrows really doesn't explain anything to me. It's stuff like this that makes people hate math.

I think we should start posting stuff about Ivanov in this thread. That'll teach these damn interlekshulls.

That's why the strength of Go programs on the biggest board (yielding the most possibilities) is still behind humans contrary to chess programs. The human brain is still better capable of picking the essential out of seamingly endless possibilities than programs. I think this has something to do with picture recognition in which humans are still better than machines (see 'captchas'). Picture recognition is also about picking the essential feature of a seamingly complicated and fuzzy thing.

The image shows that there's a way to pair up the natural numbers (1, 2, 3, ...) with the rational numbers (numbers that can be expressed as fractions, including trivial ones like 1/1) so that every natural number is associated with a rational number, and every rational number with a natural number. The initial pairings from that image are:

(1, 1/1)
(2, 2/1)
(3, 1/2)
(4, 1/3)
(5, 3/1) [skipped (2/2) because it's the same as 1/1, thus in parentheses]
(6, 4/1)
(7, 3/2)
(8, 2/3)
(9, 1/4)
(10, 1/5)
...

That same process of generating the rational numbers and drawing the arrows in that manner will eventually associate any rational number with a natural number (and vice versa). Because they be can paired off in this way with nothing left over, they have the same cardinality (roughly, they are the same size). There is no way to make any kind of analogous pairing between the natural numbers and the real numbers, which include irrational numbers (like square root of 2 and pi) that can't be expressed as fractions.

Comparing Go to CAPTCHA is a terrible analogy. Go is a strategy game, CAPTCHA is a 'verification' in which you are either asked to read normal text in picture form - which computers can do as well as humans - or text that has been skewed beyond recognition, even for people with 20/20 vision.