On Apr 9, 5:03 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote: > Zuhair <zaljo...@gmail.com> writes: > > What's the proof of the following in naive set theory? > > > Not exist x. x is empty > > By Russell's paradox, there exists a set R such that R in R and R not > in R. By ex falso quodlibet, there is no set with no elements. > > -- > Aatu Koskensilta (aatu.koskensi...@uta.fi) > > Yes, your response and Smaill's are pretty much the same. Those proofs are trivial ones, they depend on Modus Ponens where from P and P->Q we infer Q, so if P is false and is a theorem then it qualifies as a step in the proof, then since P->Q is trivially true then it is a theorem, then obviously Q is a theorem whatever Q is. BUT this proof is TRIVIAL and of no importance since we already know that P is **clearly** false. I want a non trivial proof, i.e. a proof containing no step that is TRIVIALLY false. On the other hand the proof of the existence of an empty set is SHORTER it is a direct result of naive comprehension, and it contains NO trivial step as far as I can see. Not only that the proof about non existence of the empty set can be used to prove the contrary result or any theorem, which is of no importance, while the proof of existence of the empty set is not shared with any other theorem proved, so it is a genuine proof. One can easily see that the proof by principle of explosion is not Content- full, i.e. not related specifically to the result it proves.
This calls for a re-definition of what a *Proof* is. But I don't know if this is possible.