Does chess yield the largest meaningful finite number?

I simply mean all numbers.

It is basic calculus, nothing special, but the conclusion is. It is the situation that two valid rules to compare numbers lead to two irreconcilable conclusions.

I think that you agree that all numbers above 1 are reflected by all numbers between 0 and 1, don't you?

The same time do you agree that all numbers between 0 and 1 have their counterparts between 1 and 2 and 4 and 5 for instance?

Those two statements bite each other, yet you can not say that one is true and the other is not. Which one is true? And which one is false? It is the combination which is a paradox and gives insight into the abyss of infinity.

I don't get loek's posts in fact. Given a proper definition of what bigger means, we can get all sorts of cool results (like Q being the same size as N, but smaller than R), without getting to any paradox. You have to be careful about what "bigger" means though.

There is no paradox here. I think you consider the cardinal of R (or [0,1], or the set of numbers above 1) to be an integer (in N), which not the case. we should be extra careful when talking about the number of elements in a set since it is generally not a "number" as we generaly mean it. There is an infinity of elements in [0,1] there is the same infinity of numbers in [1,infinity[, and the same infinity of numbers in [0,3] or [1,2]. There is no problem here.

Grahams number to the power of grahams number is the largest :)

edit:

In fact, instead of using 3, replace each 3 with Grahams number G. The take this new G', and use to build another one....

That's... really weird.  Over the last 30 minutes I started typing... then erasing different ways to resolve it.  I see what you're saying now.  I'll think about it after I'm rested for fun, but as you said definitely seems like a paradox.  It suggests to me our terms aren't as well defined as we think.

edit... time to sleep...

This is precisely what aleph numbers represent.

Aleph zero is the cardinality (i.e. the "number of elements") of a countably infinite set. To count all natural numbers, you proceed like this:

0, 1, 2, 3, 4, 5, 6...

In this way you could systematically and progressively get through all of them. To count all integers, you proceed like this:

0, 1, -1, 2, -2, 3, -3...

To count all rationals, we need to come up with a more clever method. As mentioned above, a set like Q is really just a N x N cartesian product - thus, we can imagine Q as all the points on a complex plane where both the real and the imaginary part are integral (except for those where the imaginary part, which would be agreed to represent the denominator, is zero). With that, all we need to do is devise a systematic path to traverse all the points sequentially and without repetition. One such sequence (the pattern in it might not be obvious, but there is one) is:

0/1, -0/1, 1/1, -1/1, 0/2, -0/2, 0/3, -0/3, 1/2, -1/2, 2/1, -2/1, 3/1, -3/1, 2/2, -2/2...

(There are practical repetitions, as all the numbers with 0 in the nominator amount to 0, and all numbers which are equal to one another, such as 2/2 and 3/3, are "dupes" as well, but this is not important - what matters is that we won't "miss" anything along the way.)

When we turn to R, the set of real numbers, we will find that coming up with a similar pattern to sequentially traverse the whole set is impossible. Here, we have an infinite number of irrational numbers everywhere in between the rational numbers, and as they cannot be expressed as fractions (and many of them, being transcendental, cannot even be expressed as zeros of any polynomials with rational coefficients), we have no means to "label" them consistently. While the sets covered above, i.e. N, Z (integers), and Q can be represented in a discrete way (meaning that their elements can be ordered sequentially one after another), the set R is continuuous.

The cardinality of the continuum, (R, the set of reals, is a continuum set) is 2^(aleph zero). The continuum hypothesis states that it also represents aleph one, the next cardinal number after aleph zero. If this hypothesis is true (it has not been proven as true or false to this day), it would mean that there are no infinite sets that are larger than a countable infinite set (N, Z, Q) but smaller than an uncountable infinite set such as R.

Great post, really well explained.

My point is that aleph numbers do not behave like "normal" numbers at all. That is why we should be careful with them. The paradox loek gets into is, as far as I understand, a consequence of the fact that aleph numbers cannot be manipulated the same way we do with integers.

About the continuum hypothesis. I am pretty sure it has been proven undecidable (which means the theory we generally use to do maths, called ZFC, is not powerful enough to answer that question).

Define meaningful?

Because some people play Cardinal Chess, with the Cardinal on a 10 x 8 board.

If that IS meaningful, then its a bigger number.

Yes, something like this is what I was thinking.

Bear with me because this is a new subject for me :)

edit... this was all wrong, sigh.  Never mind.

logging off this time for sure lol

Yes, as a consequence of Gödels findings the continuum hypothesis is actually considered undecidable from within the Zermelo-Fraenkel system.

I have not stumbled across a single point of failure in Loek's reasoning, however there must be some inconsistencies. In particular, it is not clear to me at all how he came to consider the amount of elements in the set or reals to be a natural (i.e. finite) number. As explained above, the number of elements in R is actually 2 to the power of the amount of the elements in N (or Z, Q, or any countably infinite set), i.e. 2^(aleph zero).

A Mathematical Mystery Tour (a BBC popular science documentary from 1985) does briefly discuss these matters - start around the 25:00 mark (chess receives a mention as well, from 36:00).

If it is finite.

It is not meaningful.

What do I mean?

If you love something - You'll never end it.

Okay I may have got it wrong then. I don't see clearly what the paradox is. I just wanted to emphasize aleph numbers are not standard numbers, but it turns out it is not what loek was talking about.

Loek, could you explain what the paradox is ? Or jaaas, if you understood, can you tell me?

@tronchenbiais: I don't know if I can explain it in another way then done previously. I will try to use a different perspective:

I start with the idea that every number is unique and when it is identified different from any other number. That makes every number, even all real numbers, countable. I know that R implies uncountable, but to get a grasp on the feeling of infinity, I leave that out of the equation for the moment.

My starting point is the situation in which every number is countable. And then I start to get in my loop of reasoning. That loop has to go on forever and ever. It can never stop as every level of measurement is unlayering another one.

A paradox has the characteristic that it never stops. The moment you acknowledge version 1, version 2 becomes true and the moment you acknowledge version 2, version 1 becomes true.

It is the same here:

the moment you acknowledge that there are x numbers between 0 and 1, you have the same number of numbers between 4 and 5.

The moment you acknowledge the same number of numbers between 1 and infinity for each different unit, you have that same amount of numbers mirrored between 0 and 1.

An example:

there are 10 numbers, hence 10 mirrored numbers between 0 and 1.

That implies that there are also 10 numbers between 1 and 2 and 2 and 3, hence instead of 10 numbers, we now have 100 numbers, having each 100 numbers in between.

But hey, all those 100 numbers and their subnumbers are mirrored between 0 and 1, hence we have also 10000 numbers between ....

But hey, all those 10000 numbers between 0 and 1 can also be found between 2 and 3 and 4 and 5, hence ...

This is the endless accelerated swinging I was trying to explain. I call it a paradox, because you can never stop at one definition and you have to return to the other definition.

Ok I get it. It's funny. It's true that if you don't know about the 'different kind of infinities', It is very puzzling.

This.

Headless chickens running around in circles.

This entire thread has no purpose because it has no value.

Get a better thread.

Whoaa so much math haha this is awesome :D

There are something like 10^43 possible chess positions (upper bound=10^46.7). Game tree analysis yields 10^123 possibilities, but that is an 80-move long game with a branching factor of 35. Length can be improved substantially without affecting branch factor.

For comparison, there are about 10^81 atoms in the universe.

But then, what makes the game tree analysis of chess meaningful?

Of course not, take for example Capablanca chess 10x10 and you immediately get a higher number. Sorry, but the OP question is stupid.