A paradox is essentially something that cannot be handled by logic from within your axioms. A simple, standard one is:
The barber shaves all men and only those men who do not shave themselves. Who shaves the barber?
If the barber shaves himself, he shouldn't. If he doesn't, he should. bad explanation?
Originally posted by royalchickenNow, if you would be so kind, apply that to my original post so that I can understand what you mean.
A paradox is essentially something that cannot be handled by logic from within your axioms. A simple, standard one is:
The barber shaves all men and only those men who do not shave themselves. Who shaves the barber?
If the barber shaves himself, he shouldn't. If he doesn't, he should. bad explanation?
Look at the function f(x):=1 for all x
If you use a simmilar reasoning as in the problem '1=2' you'll get that because f(a)=1=f(b) for all a and b, a = b.
The problem is that the inverse of g(x)=x^2 is a metafunction, meaning that g^-1(x) is not one point, but a set of points. g^-1(x)={sqrt(x), -sqrt(x)}. So is sin^-1(0)={n*pi | n an integer}. Because metafunctions are more difficultto handle (they don't have a derivitive, for example), and contain more information than you need, it is easier to define only the inverse of a funtion on an interval: sqrt(x) is the inverse of x^2, x in [0,inf], arcsin(x) is the inverse of sin(x), x in [-1/2*pi, 1/2*pi].