MHB Proving Independence of Axioms in Game Theory: A Case Study

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The discussion centers on proving the independence of four axioms related to game theory. Axiom 1 states that each game involves two distinct teams, while Axiom 2 specifies a minimum of four teams. Axiom 3 asserts that exactly six games are played, and Axiom 4 indicates that each team plays once against the same opponent. The user has established the independence of Axioms 3 and 4 but seeks assistance in justifying Axioms 1 and 2. Additionally, the user is tasked with formulating three theorems derived from these axioms, with one theorem already proposed regarding the number of games played by teams.
crobertson0308
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I have four axioms and I am stuck trying to prove the independence of these axioms.

Axiom 1: Each game is played by two distinct teams.
Axiom 2: There are at least four teams.
Axiom 3: Exactly six games are played.
Axiom 4: Each distinct team played once against the same team.

I've justified both Ax 4 and 3 are independent but need help justifying the other two axioms
 
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Hi,
Here is a set of models that prove the independence. I leave it to you to verify that each is actually a model.

2csausi.png
 
Would the undefined terms be elements(teams, game)...relation(is, are)?
 
I need to create three theorems that follow from the four axioms. One theorem I came up with was if there are exactly four teams, then each team plays exactly three games. I'm having trouble coming up with another two. The only path I'm seeing is increasing the number of teams and seeing what happens with the models.
 
msalamon said:
I need to create three theorems that follow from the four axioms. One theorem I came up with was if there are exactly four teams, then each team plays exactly three games. I'm having trouble coming up with another two. The only path I'm seeing is increasing the number of teams and seeing what happens with the models.

What did you get as the undefined terms?
 
For the objects: game(s), team(s)
For the relations: is, are, and I wasn't sure of played should be considered a relation or not

- - - Updated - - -

Pardon my typo; of should be if.
 

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