Creating theorems for an axiomatic system

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In summary, there are four axioms in this conversation regarding the number of teams and games played. The first axiom states that each game is played by two distinct teams, while the second axiom states that there are at least four teams. The third axiom states that there are at least six games played, and the fourth axiom states that each team can play at most four games. Using these axioms, a theorem has been proposed that if there are exactly four teams, then there are at most eight games. In order to come up with two more theorems, the general idea is to fix one variable and determine limits on the others. For example, if there are exactly six games played, how many teams can there be? Or, if
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malloryjohn
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I have the axioms:

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 played
Axiom 4: Each team played at most 4 games.

And I have come up with the theorem: If there are exactly 4 teams, then there are at most 8 games.

I need to come up with two more theorems. I'm pretty stuck on where to go from here, so any advice would be greatly appreciate
 
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How many teams can there be, if there are exactly six games played?

If each team plays only once, how many teams can there be (if only teams that play games count)?

The general idea is: fix ONE of your variables, and determine limits on the others.
 

FAQ: Creating theorems for an axiomatic system

What is an axiomatic system?

An axiomatic system is a set of axioms (fundamental assumptions) and rules of inference that are used to prove theorems in a certain field of mathematics. It serves as the foundation for logical reasoning and is essential in creating mathematical proofs.

How do you create theorems for an axiomatic system?

To create theorems for an axiomatic system, you must first understand the axioms and rules of inference that make up the system. Then, you can use logical reasoning and deductive logic to build upon these foundational elements and prove new statements or theorems.

What makes a theorem valid in an axiomatic system?

A theorem is considered valid in an axiomatic system if it can be logically derived from the axioms and rules of inference within the system. This means that the theorem is a true statement based on the assumptions and logical reasoning used in the system.

Can you provide an example of creating a theorem for an axiomatic system?

For example, in Euclidean geometry, one of the axioms states that a straight line can be drawn between any two points. Using this axiom and the rules of inference, we can prove the theorem that states two parallel lines will never intersect.

Are there any limitations to creating theorems in an axiomatic system?

Yes, there are limitations to creating theorems in an axiomatic system. The theorems must be based on the axioms and rules of inference, and any statement that cannot be logically derived from these elements cannot be proven within the system. Additionally, axiomatic systems are limited to a specific field of mathematics, so the theorems created within the system may not be applicable to other fields.

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