MHB Creating theorems from the given axioms

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The discussion focuses on creating theorems based on provided axioms related to team games. The user has established two theorems, with the first being proven and the second needing further proof. The user seeks assistance in proving the second theorem and generating a third theorem. The community encourages sharing progress to facilitate effective help. The conversation emphasizes collaborative problem-solving in mathematical theorem creation.
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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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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?
 

Thread 'Deductive proof in logic formal systems'

I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
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