Creating theorems from the given axioms

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In summary, the conversation discusses the creation and proof of three theorems based on four given axioms. The first theorem states that there are at least two teams that play a game, while the second theorem states that if there are exactly four teams, then there are at most eight games. The third theorem has not yet been created or proven, but assistance is being sought.
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dezzi1
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Given the following axioms, create 3 theorems.

Axiom 1: Each game is played by two distinct teams.
Axiom 2: There are at least four teams.
Axiom 3: At least six games are played.
Axiom 4: Each team plays at most four games.

I have already proven each axioms independence.

These are the following theorems I have:
1. There are at least 2 teams that play a game
2. If there are exactly 4 teams then there at most 8 games

I have completely proven theorem 1. I just need help proving theorem 2 and creating/proving a third theorem.

Any help will be greatly appreciated!
 
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  • #2
Hello and welcome to MHB, dezzi! :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?
 

1. How do you create theorems from the given axioms?

To create theorems from given axioms, you must use logical reasoning and deduction. This involves examining the axioms and their relationships, and using rules of inference to derive new statements, known as theorems.

2. What are the key principles for creating theorems from axioms?

The key principles for creating theorems from axioms include understanding the logical structure of the axioms, using valid rules of inference, and making sure that the theorems are logically consistent with the axioms.

3. Can any theorem be derived from a set of axioms?

No, not all theorems can be derived from a set of axioms. The axioms must contain enough information and be logically consistent in order for theorems to be derived from them. If the axioms are too limited or contradictory, certain theorems may not be able to be derived.

4. How do you know if a theorem is valid?

A theorem is considered valid if it can be logically derived from the given axioms using valid rules of inference. This means that the theorem is a true statement based on the axioms and logical reasoning.

5. Can a theorem be proven wrong?

No, a theorem cannot be proven wrong if it is derived from valid axioms and logical reasoning. However, it is possible for the axioms themselves to be proven wrong or for new information to be discovered that renders the theorem no longer applicable.

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