They are true without proving.

But something very very interesting:

Guedel proof that there doesn't exist a full set of axioms, e.a. there exist a statement that cannot be proved using finite number of axioms.

But I think I've demonstraighted that some things cannot be proven technically, but are so obvious they don't need to be. In my oppinion, the critical number theorem is about as obvious as 2 + 2 = 4.

Ok not quite as obvious, but think about it, if f(a) and f(b) are not the maximums and/or minimums then there is to be a point somewhere between them thats higher or lower then they are. If its higher then it has to turn around to come back down, the turning points of a function are the critical numbers. Nothing to prove in my mind.

]]>2+2 = 4 is an identity. It is also a proof in and of itself, because it is evident from the rules of addition that 2+2 = 4; IOW, no further steps are required to show that it is true.

OK, Ok, ok, I'll do it:

Assert: 2+2 = 4.

By the rules of arithmetic, 2+2 = 4.

Substitute: 4 = 4

Therefore, 2+2 = 4.

Woohoo!

]]>I agree proofs should be done most of the time, but sometimes proving an obvious concept feels redundant.

]]>Perhaps the proof is not difficult, but, because it's a concept that's simply and intuitively grasped, it is not necessary to the text, and therefore beyond its scope.

]]>To me it seems obvious. Why would we have to prove it?

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