How many possible outcomes are available? Why is the number of permutations finite? Is it solely due to the finite starting positions, or do the choices involved include factors of infinity?
- Think universe = big chess board, extremely large but finite, choices are not fully available, resulting in the final stages of the universe. Is there one outcome, or a finite yet extremely large number of possibilities?|||Yes.
As all moves/outcomes are discrete, that is a constraint.
The number of moves may be potentially infinite, but I don't think that's what you're asking.
The Universe is not like a chess board as it is not deterministic or fully knowable, so the estimation of future outcomes can only be based on probabilities and statistics, not discrete outcomes. However, as causality is emergent from quantum non-determinism, we can maybe approximate it via cause and effect.|||All possible states of a chess game are knowable at all times. Conversely Heisenberg uncertainty means that the state of the Universe is not completely knowable ever. Indeed it could be argued that quantum existence is 'fuzzy', not discrete.
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|||Because a chessboard is limited in size and there are restrictions on how pieces may move, there are a finite number of outcomes, but it's still a very large number. Nobody could possibly list them all. Some have argued that the number may very well be larger than the number of individual atoms in the observable universe. As such, it's only practical to treat the game as if it had an unlimited (albeit not necessarily infinite) number of moves regardless of whether or not we count the "50 Move Draw" rule.
Each move by each player reduces the number of possible permutations remaining. In other words, each move by each player influences what moves are still available. The goal is to end with a permutation that fits the victory conditions of the game.
One might argue that a human life is the same way: every choice made cuts down on the number of possibilities left to explore, and the choices that each of us make end up impacting each other's possibilities. This was the view the Jean-Paul Sartre expressed, and it's what he was talking about when he declared that "Hell is other people." The only difference, really, is that there is no permutation that guarantees a "win" condition in human life. There are only permutations upon permutations, and then you stop.
As for the universe, it might be the same way, except that it doesn't seem to be the case that the universe thinks or makes decisions...it's just a collection of stuff, nor do we know of any beings who alter the entire course of reality based on decisions they make. As such, there is only one final permutation for the universe beyond the control of anyone or anything (but if there are other universes--an entire multiverse--there could very well be an unlimited number of permutations).|||The number of possible legal chess games is larger than the number of atoms in the observable universe.
A finite number which is too large ever to be calculated is often not distinguishable from an infinite number (a problem which cannot be solved during the available age of the universe is an unsolvable problem).|||It is finite on the chess board, yes. But the number of permutations is extremely large, which is why even computers can't plan all the way through a game but only a certain number of steps ahead.|||I'd be guessing if I gave an answer to the universe part of the question...|||Do you mean zeros move?
firstly you can only move 10 figures. Latter more then less (you are "killed")
Number of stances are finite:
for concrete number you can define function from position on chest board to {0,...10} where 1 means white pawn... Number of white pawns can be 0...8 and there's 1. coordinate is greater then 1. one hunter can only be on white fields and that is it. To calculate it it will take some time.
I will just gave you upper limit of that number:
If we assume every position is legal and number of figures of same type can vary to 64 then (using upper defined model) number of stances are 11^64
Number of games:
If we move horse "up" and down repeatedly it is clear that there is infinite number of games. But that "strategy" is joke, so we will ban that kind of movement.
Def. (S,%26lt;) is strategy if it is totally ordered set. (we assume that player will not say gee, I don't know what is better situation)
As you can see: while we know what is preferred stance for player 1 we don't know whether his preferred move is legal, making this result of upper bound ridiculously larger.
Number of pair of strategies is ((11^64)!)^2 and that pair is enough to define the course of our game (note that in all strategies there will also be something like "I prefer to be checkmated"). Badly using Stirling's approximation number is lower then (11^64)^(11^64)^2 I believe that concrete number is about 10^123 ( http://en.wikipedia.org/wiki/Shannon_num鈥?/a> )
Judging by continuous debranching of atoms (greek undividable) it is very bold to say that universe isn't continuous and is bounded. If you mean to say that even number of "games" are finite that would mean that you also assume that time is always natural number smaller then "end of time". In book "infinity and mind" i see writer sounding like infinitist fundi, while he have some good arguments.
My guess is 2^C and number of games is also ( (2^C)^(2^C)=2^C ). Joke strategies increase number of moves.
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