toshiaki wrote: > "David Marcus" <DavidMarcus@alumdotmit.edu> wrote in message > news:MPG.firstname.lastname@example.org... > > toshiaki wrote: > > > My question is that why selfreferential statement is permited as theorem of > > > mathematics. > > > > Why shouldn't it be? > I cannot judge about it now. > When we use the liar's statement in ordinary conversation, it only leads to > nonsence contradiction.
Luckily, math isn't the same as ordinary conversation. The best explanation I've seen of the Liar's Paradox was in the journal Mathematical Intelligencer. Basically, the author showed that the Liar's Paradox is analogous to assuming that an equation without a solution has a solution.
> Though ZF excluded Russel's paradox and other paradoxes from its system, why > doesn't it inhibit > to use this statement to prove the incompleteness theorem?
The proof of the incompleteness theorem has some similarity to the Liar's Paradox, but it isn't the same. Instead of saying "I am not true", you only say, "I am not provable". If provable statements must be true, you can conclude that you are true.
> I'm afraid that this is nonsensical question?
More sensible than some of the other things you've said.