11. Evolution of Cantor’s Middle Third Set to the 4th level
Next we make an amazing observation. Bytaking the ternary expansion of any number c in C and replacing each instance of2 by 1, we obtain the binary expansion of some number c'in the unit interval. This gives a one-to-one correspondence of C with the set of all numbers in I(written in binary). It follows that the cardinality of C is the same as that of I, and since the latter is an uncountable set(by Cantor’s Diagonal Argument), it follows that the Cantor Middle Third Set is not only infinite but uncountable.