Infinities := means is defined to equal.

phoenixthoth

infinities := means "is defined to equal."

:= means "is defined to equal."

x_y means y is a subscript of x.

N:={0, 1, 2, ...}

[P,0](S):=S
[P,1](S):={x : x is a subset of S}
for n >= 1,
[P,n+1](S):={x: x is a subset of [P,n](S)}

card(S) is the cardinal number of S.

1_n:=[P,n](N).

aleph_n := card(1_n).

2_1:={x: x is an element of 1_n for some n in N}; i.e., 2_1 is the union of {[P,0](N), [P,1](N), [P,2](N), ...}.

what is card(2_1)?

for n>=1,
2_(n+1):={x: x is an element of [P,k](2_n) for some k in N}; i.e., 2_(n+1) is the union of
{[P,0](2_n), [P,1](2_n), [P,2](2_n), ...}.

what is card(2_n)?

3_1:={x: x is an element of 2_n for some n in N}.
for n>=1,
3_(n+1):={x: x is an element of [P,k](3_n) for some k in N}.

what is card(3_n)?

for m>=1,
(m+1)_1:={x: x is an element of m_n for some n in N}
for n>=1,
(m+1)_(n+1):={x: x is an element of [P,k](m+1_n) for some k in N}.

what is card(m_n)?

:={x: x is an element of m_n for some m,n in N}.
for n>=1,
[n+1]:={x: x is an element of [P,k]([n]) for some k in N}.

what is card([n])?

[N,1]:={[n]: n is an element of N}.
for k>=1,
[N,k+1]:={[n]: n is an element of [P,j][N,k] for some j in N}.

what is card([N,k])?

Last edited by a moderator:

Re: infinities

Originally posted by phoenixthoth
:= means "is defined to equal."

x_y means y is a subscript of x.

N:={0, 1, 2, ...}

[P,0](S):=S
[P,1](S):={x : x is a subset of S}
for n >= 1,
[P,n+1](S):={x: x is a subset of [P,n](S)}

card(S) is the cardinal number of S.

1_n:=[P,n](N).

aleph_n := card(1_n).

2_1:={x: x is an element of 1_n for some n in N}; i.e., 2_1 is the union of {[P,0](N), [P,1](N), [P,2](N), ...}.

what is card(2_1)?

for n>=1,
2_(n+1):={x: x is an element of [P,k](2_n) for some k in N}; i.e., 2_(n+1) is the union of
{[P,0](2_n), [P,1](2_n), [P,2](2_n), ...}.

what is card(2_n)?

3_1:={x: x is an element of 2_n for some n in N}.
for n>=1,
3_(n+1):={x: x is an element of [P,k](3_n) for some k in N}.

what is card(3_n)?

for m>=1,
(m+1)_1:={x: x is an element of m_n for some n in N}
for n>=1,
(m+1)_(n+1):={x: x is an element of [P,k](m+1_n) for some k in N}.

what is card(m_n)?

:={x: x is an element of m_n for some m,n in N}.
for n>=1,
[n+1]:={x: x is an element of [P,k]([n]) for some k in N}.

what is card([n])?

[N,1]:={[n]: n is an element of N}.
for k>=1,
[N,k+1]:={[n]: n is an element of [P,j][N,k] for some j in N}.

what is card([N,k])?
Good evening phoenixthoth
I wish i could answer you, maybe in my next life when i am a math wiz. I will leave that to someone else.

Could you calculate me this> Is the maximum amount of moves in a chess game finite? If so how many. The board is limited to 64 spaces that are finite, there are 32 pieces that are finite plus new pieces that are finite and last but not least the amount of moves each piece could make in relation to the others. Can you get infinity from fixed variables? or is there a finite number of games?

phoenixthoth

i've been thinking about this question, or a related question, on and off, for a while now. the question i was thinking about is how many distinct legal chess games are there in which each player made any possible legal move and what percentage of those games are draws? one game would be considered distinct from another if, for example, a bishop was moved two spaces instead of one.

the wording is slightly different, but tell me if our questions are the same.

i just realized that the answer to my question is infinite because the knight (and most pieces) can cycle back and forth between two positions indefinitely.

i guess a related question is how many different games are there if infinite cycling is prohibited? the rub is that sometimes infinite cycling is inevitable though these are when draws occur. so i guess the question could be rephrased to ask what percentage of chess games don't end in infinite cycling?

***

what i'm trying to get at with the original post is that there is a cardinal bigger than all the alephs. from this cardinal, a list of bigger cardinals can be constructed. then there is a cardinal bigger than all those on that list. and so on forever. my ultimate goal is to try to find something bigger than all cardinals but that's impossible.

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Hurkyl

Staff Emeritus
Gold Member
You forget the triple repetition rule; if a game state is repeated 3 times, the game is a draw. Since there are only a finite number of game states, this ensures a game ends after a finite number of moves.

phoenixthoth

if (x y) is the binomial coefficient then an upper bound should be in the ballpark of Sum( (64 x) , {x,2,32} ) which is in the ballpark of 10^43 - 10^44.

Originally posted by Hurkyl
You forget the triple repetition rule; if a game state is repeated 3 times, the game is a draw. Since there are only a finite number of game states, this ensures a game ends after a finite number of moves.
Also there is a finite number of good moves of which if you are playing against a no mistake player, you will not make a second move.

Originally posted by phoenixthoth
if (x y) is the binomial coefficient then an upper bound should be in the ballpark of Sum( (64 x) , {x,2,32} ) which is in the ballpark of 10^43 - 10^44.
So we found something interesting tonight, the amount of games may be the plank length 10^43

i just realized that the answer to my question is infinite because the knight (and most pieces) can cycle back and forth between two positions indefinitely.
I think you may have to consider this> Also there is a finite number of good moves of which if you are playing against a no mistake player, you will not make a second move. If you use that variable a perfect player, the amount of games have to be finite.

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