Proving Independence of Axioms in Game Theory: A Case Study

In summary, the conversation was about proving the independence of four axioms related to game theory. The axioms state that each game is played by two distinct teams, there are at least four teams, exactly six games are played, and each distinct team played once against the same team. The person has justified the independence of two of these axioms and is seeking help to justify the other two. They have created one theorem stating that if there are exactly four teams, each team plays exactly three games, but are struggling to come up with two more theorems. The conversation also briefly mentions the undefined terms and relations used in the axioms.
  • #1
crobertson0308
1
0
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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  • #2
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
 
  • #3
Would the undefined terms be elements(teams, game)...relation(is, are)?
 
  • #4
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.
 
  • #5
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?
 
  • #6
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.
 

Related to Proving Independence of Axioms in Game Theory: A Case Study

What is the concept of "Independence of Axioms" in science?

The independence of axioms refers to the idea that certain fundamental principles or assumptions in a particular scientific theory or system do not rely on or can be proven by other principles within the same system. This means that these axioms are accepted as true without requiring further justification or proof.

How is the independence of axioms determined in science?

The independence of axioms is determined through rigorous testing and experimentation. Scientists use empirical evidence and logical reasoning to determine the validity and consistency of a set of axioms. If a particular axiom can be derived from other axioms, it is considered dependent and not independent.

What is the significance of independence of axioms in scientific theories?

The independence of axioms is crucial in the development and acceptance of scientific theories. It ensures that the fundamental principles of a theory are sound and consistent, and helps to prevent circular reasoning. It also allows for the expansion and refinement of a theory by adding new axioms that are independent of the existing ones.

Can the independence of axioms change over time?

Yes, the independence of axioms can change as new evidence and knowledge is discovered. What may be considered an independent axiom in one scientific theory or system may become dependent in another. This is because as our understanding of the natural world evolves, our fundamental assumptions about it may also change.

Are there any limitations to the concept of independence of axioms in science?

While the concept of independence of axioms is widely accepted in science, it is not without its limitations. Some argue that it is impossible to prove the independence of all axioms, and that there may be hidden assumptions or biases that influence our understanding and acceptance of certain axioms. Additionally, the concept of independence of axioms may not apply in non-scientific fields such as philosophy or mathematics, where different methods of reasoning are used.

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