Does chess yield the largest meaningful finite number?

First of all: Go has more possibilities than chess. So, the question to the thread is definitely: NO.

Second, Graham number is a number. In set theory one has the Aleph as a size of infinite sets. The smallest infinite number is Aleph0 and marks the border between finite and infinite numbers. (In a paradoxical way.) You also have Aleph1, which is the size of all countable ordinal numbers. (Without that can you not speak of any countable number.) That has a meaningful infinite number, hasn't it? And because it restricts all countable ordinal numbers is it directly related to all finite numbers. The relation between Aleph0 and Aleph1 is that Aleph1 is 2 to the power of Aleph0.

Can you imagine a number that is the 2 to the power of an infinite number? Now that is huge, while Graham Number is always infinitesly smaller then Aleph0.

For more information: http://en.wikipedia.org/wiki/Aleph_number

That's really interesting... with a lot of odd concepts to grasp at once.

I have no idea why it's considered useful to define a smallest infinity.  Furthermore the definition being log2(Aleph1) seems completely arbitrary both for its size and for being expressed with algebraic operations.

@wafflemaster: the smalles infinite number seems a paradox to me as well. I always thought that there was only one infinite number, because you can't make any further distinctions. All other numbers are at least one smaller, hence finite.

bla bla bla WHO CARES

Well, aleph zero is actually the cardinal number of all countable sets (naturals, rationals, etc.). 2^(aleph zero) is the cardinality of the continuum (the set of real numbers being an example of), and, should the continuum hypothesis be true, it would also be aleph one.

Still, throwing this into the mix is a bit of an apples/oranges comparison. An aleph number itself (rather than its index) isn't a natural (what the OP probably wanted to restrict himself to) or a even a real number, thus it certainly cannot be treated as a "finite number" (as per the thread title). In order to be able to incorporate this concept, we would need to extend into surreal numbers. And even then, what is found at various "levels of infinity" still isn't really directly comparable to what is commonly understood as "numbers", i.e. naturals, integers, rationals, reals, etc.

The only interesting part of this thread is that nobody pays attention to LetsGoHome . He is a dick , doesn't understand the subject and his grammar is awful. His only goal is to insult. Having a low chess-rating has nothing to do with having some mathematical insight.

Yeah, I mean, what happens when you subtract or divide the so called smallest infinite number right?  I suppose they'd say its cardinality does not become smaller.  But then why can you use infinite multiplication of 2s to transfer between cardinalities?

Oh wait... that actually sort of makes sense lol.  The difference between a countable and uncountable set is to simply add or remove an "order" or "level" of infinity.

Anyway, I understand mathematicians only define things like this because it's useful for something else.  So I'm willing to accept concepts like "smallest infinite" but until I know how to do something useful with it it certainly seems like day dreaming or fiction.

You can devise (more or less trivial) methods to progressively count through all of the elements of countable sets such as N (the set of all natural numbers) or Q (the set of all rational numbers, which is a N x N cartesian product).

However, you cannot order the elements of R (the set of all real numbers) in any such way as to be able to count through them systematically and sequentially while making tangible progress in the process. As such, even though N, Q, and R are all "infinite", R must be larger than N or Q.

This is why it's odd to me they chose to include an algebraic expression when talking about these type of concepts.  As you said (and I'm guessing) these aren't numbers as we normally think of them but could better be described as concepts.

Isn't that funny though?  It almost gets philosophical more than mathematical.

To say what jaaas said in another way, for a countable infinity, you can literally count them "1, 2, 3, 4..." even though you'll never finish.

For the real numbers, you can't count them. If you start with 1, can you tell me what the "next" real number is? If you say 1.001, I'll say 1.0005 is right between them. You can always find another number between number a and number b, so you can't ever count them in the same way. That's why the reals have a higher order of infinity than the natural numbers. 

That's the only explanation I've ever been able to use to get anyone to really understand why one infinity can be bigger than another.

As for Graham's number...it's really fun to try to comprehend, and to learn Knuth's up-arrow notation. As an operation, it's like a progression of exponentiation. (Multiplication is a progression of addition, exponentiation is a progression of multiplication, up-arrow is a progression of exponentiation in this sense). For those wondering, it's not possible in our universe to write this number, even in scientific notation. Not only that, it's not even possible to write the number of digits that would be in the exponent of the scientific notation, in scientific notation. I'm pretty sure you can keep saying that for quite some time before you get to something that's possible. 

@jaaas: of course can you not order them. The point that you reach the distinction between finite and infinite isn't located anywhere. It is the feeling sitting on an eternal accelerating train on its way to infinity.

And no, R is not bigger than Q or N, because the number of elements in R is an N-number. Let's say that there are x numbers between 0 and 1. That same number exist between 1 and 2 and -3 and -4 obviously. The number of numbers between 0 and 1 is equal to all numbers bigger then 1.

Therefor are there an equal amount of numbers between 1 and 2 as there are numbers bigger then 1. (all numbers between 0 and 1 can have 1 added)

In the end you have a huge number of numbers. Very infinite, yet very accountable and an element of N. That number is of course bigger then all other numbers added together and multiplied by two (all negative numbers as well, you know). But you can't reach that number without having an ordered set of numbers to that number. Hence, the moment you reach one possible answer to the question how many numbers exist in R, you have yourself already set to take the train to the next station of infinity. It will go on for ever and ever. Proving that R is bigger and smaller than N or Q at the same time.

Assuming there are on average M possible moves in a position , and assuming a game takes D plies to finish, then there are P=M^D possible positions. Some of these can be reached by transpositions , thus P will be even less if we eliminate doublures.

Now say M=40 and D=300 ( a long game) then P=40^300 . Quite big , but nothing compared to the Planck dimensions grid.

There are really interesting discussions about infinity here, but I am just going to give a not so clever argument:

Let N be the highest meaningfull number. Then N+1 would be "the first (smallest) number that is so high it can't be meaningfull". Wouldn't that make N+1 a meaningful number ?

My point is that the notion of meaningful is vague. What makes a number meaningful ? Or in other terms, what does that mean to mean something ?

That can't be true by your own definition... because the number of numbers between 0 and 1 occurs more than once between 1 and... 5.

Unless I'm thinking of "equal" in a way you didn't mean it.  Do you mean their cardinality is the same?

What?

@wafflemaster: my reasoning was to show that when you start thinking about the logic of finite and infinity you will end up in paradoxes.

From one point of view it is logic that there are an equal amount of numbers between 0 and 1 and any other number. In the same time we have this function 1/x, which mirrors all numbers above 1 in between 0 and 1. Hence are all numbers reflected within this little difference. Implying that the number of numbers in between 0 and 1 is just as big as all the numbers above 1.

Furthermore is it clear that to every number between 0 and 1 can you add 1, 2, 3, 4 etc. That shows that the first statement that there are an equal amount of numbers between two numbers. Which are again all mirrored between 0 and 1. Furthermore....

That is what I mean with the ever accelerating train. The more you dive into it, the more you get a grasp of the ever escaping determinatability of infinity (and hence finity as well).

You can not say that R is bigger than N, because the total number of elements is element of N. N can never be a member of R at that time, because there are an infinite number of numbers in between every number element of N element of Q element of R. The size of R can only be expressed with a number that is bigger than the number of elements in R itself, but that number is by definition element of N, hence N has to be bigger than R at that time.

The moment you realize it, becomes N a part of Q and R again, hence becomes Q again bigger than N and R bigger than Q and N and you can get into the train for the next line of acceleration deeper into infinity.

waffle, it is a little tricky. In terms of inclusion, [0,1] (the set of numbers between 0 and 1), is smaller than [0,5] (beceause every number in [0,1] is also in [0,5]).

However in terms of cardinality, they have the same size. In order to proove that, we have to find a way to associate each number in [0,1] with a unique number in [0,5], in a way such that each number in [0,5] is beeing associated to once and only once (I am not sure it is very clear). One way you could do this is by associating x from [0,1] to 5*x. every number y in [0,5] is being associated to y/5, and only to this number. So we defined a correspondance between [0,1] and [0,5], with 0 coresponding to 0, 1 corresponding to 5, 0.5 corresponding to 2.5, etc. This prooves that [0,1] and [0,5] have the same cardinality. Does this sound clear at all ?

Sorry, I can't follow that logic.  Maybe I'm tried (I am) but maybe you're also crazy

When you say the number of numbers above 1 do you mean the number of reals, naturals, or what?

Yes :)  I've seen a similar proof and that's a good way to explain it.

That was my point to loek.  If he means equal in terms of cardinality I'd agree... but then his point is superficially obvious so I'm guessing he meant the actual number of elements is the same... which I disagreed with.

the actual number of elements in an infinite set is something to be careful with. Are you talking about the measure of a set of reals ? 

I am not sure what you are arguing about, maybe I'd better stay distant.

I think he's saying depending on the logic you use, you can show there are more naturals than reals or there are more reals than naturals.  My last post I edited down to the part where I'm not clear on what he's saying.