Why is it "impossible" that ?
what is wrong with the following proof that 1 is the largest real number?
1 is the largest real number :
let x be the largest real number (want to prove that x=1) then
(since and x is the largest)
but is impossible so and therefore
or x(x-1)=0 so that x=0 or x=1 since 1>0 then x=1 and thus 1 is the largest real number
Oh yes, I see.
Not sure then. All that tells me is that *if* there is a largest real number, *then* it is no greater than 1. As we know there are real numbers greater than 1, it follows that there is no greatest real number. (That is, if you think you've got the greatest one, you can always find one that's even greater, i.e. by squaring it.)
Interesting little question. Nice one.
So the flaw is in the first line, when he suppose that there's a largest element in . Because we know that is a field. should lie in and if is the largest element then . Taking any number shows that assuming the existence of a greater element in is false.
Edit : Alternatively, to end a bit more formally, take any . Clearly . Also , but , which is impossible since is the greatest element and . Hence we can't suppose that there exists a greatest element in .