Originally posted by iamatiger
I'm not saying you're wrong, but where is the error in this?
Any rational number is the ratio of two naturals. Two naturals can be encoded into a real between 0 and 1 by using alternate digits of the decimal expansion of the real to represent each integer. Therefore the reals between 0 and 1 map to the rationals.
I would agree without argument that the whole set of reals is bigger than the rationals...
The reals, by virtue of being an infinite set, can be put into 1-1 correspondence with any subinterval of reals. Therefore, the reals are EXACTLY as large as any of their subintervals. Since you accept that ''all of the reals'' is larger than any set of rationals, you must also accept that any infinite set (subinterval) of reals is larger than any set of rationals.
But that does not exactly answer your question. The error in oyur argument lies in the fact that there is no 1-1 correspondence there. For example, using your method would associate 14/37 with 0.1347. However, 0.1347 = 0.13470, so this same number also associates with 140/37 and so on. Indeed, each real associates with an infinitude of rationals.
Cantor's formal proof works in a slightly similar way.