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Wouldn't there be exactly twice (or twice + 1, if you allow negative fractions) as much fractions, since fractions are represented as two positive numbers (plus a bit if you consider the sign).

(The encoding could be "represent both numbers in binary, put the denominator in the odd bits (LSB = first bit), and the numerator in the even bits" so 2/3 => 10/11 => 1110 => 14)

The thing is that you could also use this kind of logic to show that there are more natural numbers than there are natural numbers. For example, you could associate 1 with all of the numbers from 1 to one million, and still have enough numnbers 'left' to associate each of 2,3,... with distinct natural numbers above one million.

What's worse, there are an infinite number of fractions that equal each whole number.