Anyway, I asked Elroch to prove that it's correct, according to a theorem of combinatorial games theory, that chess can be solved mathematically.
He just told us that it wasn't an axiom, by the way, MEGA. I thought we were making some progress. However, you were more correct than Elroch, seemingly, because if he can't prove that it can be solved mathematically (and that proof has to be a syllogistic proof, which is what he always demands of others) then it's a axiom.
Having an axiom that chess can be solved mathematically is exactly equivalent to an axiom that states that mankind will reach other galaxies in their space exploration. Now, I'm not saying it's impossible but I strongly doubt it. The axiom is based on "mankind can travel, therefore mankind can travel to other galaxies" and is exactly equivalent to "simple combinatorial games can be solved mathematically: therefore chess can." I would require a proof, please, or you've lost the argument.
You would lose the argument since you would not have responded in kind. You required deductive proofs from tygxc but you cannot give them to defend your own far less reasonable claims.
@Optimissed, this theorem even has its own wikipedia page.
If you want a formal proof, here is one (it's half a page long after the definitions have been made):
The original paper was published in 1913 (in German), and apparently it was the first published paper on game theory. It needed an addition published in 1927 to be a truly rigorous proof for basic chess (where there are a finite number of possible positions, but games can be indefinitely long).
Interestingly, this is the same Zermelo who has his name attached to the Zermelo-Fraenkel axiomatisation of set theory that is perhaps the most popular foundation for mathematics.
[Remark: It strikes me that Zermelo's theorem can be easily generalised to a game where there is a general finite ordered set of outcomes].
Looked at it. It's an hypothesis. It may be considered by Zermelo to be an axiom and there's no syllogistic proof to support it.
Narcissism appears to continue to be a fatal obstruction to your understanding. It's a THEOREM that was PROVED in the early 20th century and which can be understood by an undergraduate. All you had to do to avoid that blunder was read what I said. Or the wikipedia article. Or the paper with the proof in it. Or any reputable source on the subject.
"Zermelo's work shows that in two-person zero-sum games with perfect information, if a player is in a winning position, then that player can always force a win no matter what strategy the other player may employ."
That is rather unfortunate, isn't it? Do you think it detracts from Zermelo's work? The commentator says that Zermelo shows that if a player is in a winning position then that player is in a winning position. There's no other interpretation.
No, it doesn't detract from the work: it is NOT the work. It is just worded in a redundant way. The key fact is the statement of the theorem itself in a general way for games that start at a general position:
For each legal chess position exactly one of the following is true:
- White has a strategy that wins against all possible play by black
- Black has a strategy that wins against all possible play by white
- Both players have a strategy that at gets at least a draw against all possible play by the other player
We then define the three classes of position as winning positions for white, winning positions for black and drawing positions.
I hope my wording helps you.
Anyway, I asked Elroch to prove that it's correct, according to a theorem of combinatorial games theory, that chess can be solved mathematically.
He just told us that it wasn't an axiom, by the way, MEGA. I thought we were making some progress. However, you were more correct than Elroch, seemingly, because if he can't prove that it can be solved mathematically (and that proof has to be a syllogistic proof, which is what he always demands of others) then it's a axiom.
Having an axiom that chess can be solved mathematically is exactly equivalent to an axiom that states that mankind will reach other galaxies in their space exploration. Now, I'm not saying it's impossible but I strongly doubt it. The axiom is based on "mankind can travel, therefore mankind can travel to other galaxies" and is exactly equivalent to "simple combinatorial games can be solved mathematically: therefore chess can." I would require a proof, please, or you've lost the argument.
You would lose the argument since you would not have responded in kind. You required deductive proofs from tygxc but you cannot give them to defend your own far less reasonable claims.
@Optimissed, this theorem even has its own wikipedia page.
If you want a formal proof, here is one (it's half a page long after the definitions have been made):
Zermelo's Theorem
The original paper was published in 1913 (in German), and apparently it was the first published paper on game theory. It needed an addition published in 1927 to be a truly rigorous proof for basic chess (where there are a finite number of possible positions, but games can be indefinitely long).
Interestingly, this is the same Zermelo who has his name attached to the Zermelo-Fraenkel axiomatisation of set theory that is perhaps the most popular foundation for mathematics.
[Remark: It strikes me that Zermelo's theorem can be easily generalised to a game where there is a general finite ordered set of outcomes].
Looked at it. It's an hypothesis. It may be considered by Zermelo to be an axiom and there's no syllogistic proof to support it.
"Zermelo's work shows that in two-person zero-sum games with perfect information, if a player is in a winning position, then that player can always force a win no matter what strategy the other player may employ."
That is rather unfortunate, isn't it? Do you think it detracts from Zermelo's work? The commentator says that Zermelo shows that if a player is in a winning position then that player is in a winning position. There's no other interpretation.