Justifying Independence of Axioms 1-4

[narledge]

I need help with the following situation---

I need to justify the independence of all four axioms using computation and/or narrative. This is what I have so far, but would appreciate any ideas and help you may have to offer.
Axiom 1: Each game is played by two distinct teams.
Axiom 2: There are at least four teams.
Axiom 3: There are at least six games that are played.
Axiom 4: Each distinct team plays each of the other teams only once.

Undefined terms: game, team, playedIf Axiom 1 did not exist, we would not know what constitutes a game. A game could be Team A. Or a game could be Team A vs. Team B vs. Team C. Axioms 2, 3, and 4 could all still be true. Therefore, Axiom 1 is independent.

If Axiom 2 did not exist, we could start with 5 teams. Again it would be possible for Axioms 1, 3, and 4 to all be true. Therefore, Axiom 2 is independent.

If Axiom 3 did not exist, it would be possible for Axioms 1, 2, and 4 to be true. Therefore, Axiom 3 is independent.

If Axiom 4 did not exist, we could start with 4 Teams, have them play each other twice, and therefore have 12 games overall. Axioms 1, 2, and 3 are still all true. Therefore, Axiom 4 is independent.This is the feedback I have received--

The spirit of the strategy, that of considering a model in which one axiom "did not exist" (more properly stated as "is false") while others are true, is a correct way to establish independence for the false axiom. Ax 1: The alternate descriptions of possible games constitute a correct way to begin the model. Additional details are needed to concretely show in what scenario "Axioms 2, 3, and 4 could all still be true." Ax 2: This model begins with five teams. This does not make Ax 2, "at least four teams," false. The model does not establish independence for Ax 2. Ax 3: No details describe how to confirm that Ax 3 is indeed false while Ax 1, 2, and 4 are all true. Please describe the concrete details in this model. Ax 4: A model in which "we... start with 4 teams" confirms the truth of Ax 2; then "have them play each other twice" confirms that Ax 4 is false; and finally "have 12 games overall" confirms that Ax 3 is true. Additional details are needed to confirm why Ax 1 is true or false

 

[mmwave]

and why Ax 2 is true or false.Thank you for your feedback. I will provide more concrete details to further justify the independence of each axiom.

Axiom 1: Each game is played by two distinct teams.
To establish the independence of this axiom, we can consider a scenario in which a game is played by three teams instead of two. In this scenario, Axiom 2 and Axiom 3 can still be true, as there are still at least four teams and at least six games being played. However, Axiom 4 would be false as one of the teams would have played two games instead of just one. This shows that Axiom 1 is independent, as it can be false while the other axioms are still true.

Axiom 2: There are at least four teams.
To further justify the independence of this axiom, we can consider a scenario in which there are only three teams. In this scenario, Axiom 1 and Axiom 3 can still be true, as each game is still played by two distinct teams and there are still at least six games being played. However, Axiom 4 would be false as each team would not have played each other only once. This confirms that Axiom 2 is independent, as it can be false while the other axioms are still true.

Axiom 3: There are at least six games that are played.
To show the independence of this axiom, we can consider a scenario in which there are only five games being played. In this scenario, Axiom 1 and Axiom 2 can still be true, as each game is still played by two distinct teams and there are still at least four teams. However, Axiom 4 would be false as each team would not have played each other only once. This demonstrates that Axiom 3 is independent, as it can be false while the other axioms are still true.

Axiom 4: Each distinct team plays each of the other teams only once.
To further establish the independence of this axiom, we can consider a scenario in which each team plays each other twice. In this scenario, Axiom 1 and Axiom 2 can still be true, as each game is still played by two distinct teams and there are still at least four teams. However, Axiom 3 would be false as there would