- #31
Dragonfall
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There is another PF member, poutsos.A, with the exact same prose as evagelos. They usually mutually agree on each other's posts. I wonder...
Proof of ΦxΦ = Φ and the special case of \emptyset\times\emptyset =\emptyset
- Context: Graduate
- Thread starter evagelos
- Start date
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Discussion Overview
The discussion centers around the proof of the statement \emptyset\times\emptyset = \emptyset, exploring various approaches to this concept within set theory. Participants debate the validity of different proof methods, including direct proofs and proofs by contradiction, while addressing the implications of the empty set's definition.
Discussion Character
- Debate/contested
- Technical explanation
- Mathematical reasoning
Main Points Raised
- Some participants assert that the empty set has no elements, making the proof straightforward.
- Others propose a proof by contradiction, suggesting that if \Phi \times \Phi is not empty, it leads to a contradiction with the definition of the empty set.
- One participant challenges the validity of the contradiction approach, stating that it lacks a foundational theorem.
- Another participant emphasizes that the empty set is defined as containing no elements, reinforcing the contradiction argument.
- Some argue that a formal proof is necessary to resolve the debate, while others believe Hall's informal proof is sufficient.
- A formal proof is presented, outlining steps that utilize axioms and theorems in set theory to demonstrate the statement.
- Concerns are raised about the tone of responses and the necessity of formality in proofs, with some advocating for a balance between rigor and clarity.
Areas of Agreement / Disagreement
Participants express differing views on the validity of Hall's proof and the necessity of formal proofs. There is no consensus on the best approach to proving the statement, and the discussion remains unresolved regarding the acceptance of various proof methods.
Contextual Notes
Some participants note that the discussion hinges on the interpretation of what constitutes a formal proof and the definitions involved in set theory, particularly concerning the empty set.
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