Plato puts the following words in Socrates' mouth in the Phaedo dialogue:


  I mean, for instance, the number three, and there are many other examples. Take the case of three; do you not think it may always be called by its own name and also be called odd, which is not the same as three? Yet the number three and the number five and half of numbers in general are so constituted, that each of them is odd though not identified with the idea of odd. And in the same way two and four and all the other series of numbers are even, each of them, though not identical with evenness. (104a-b)


The philosophical point is that there exists an Idea (or Form) called Odd and odd numbers are merely specific instances of Odd.  The are not, themselves, identical to the concept of Oddness.

But I bolded a throwaway phrase that prompts my question: Are half of all integers odd?



Plato likely did not consider one to be odd nor did he likely consider either zero or negative integers among his set of "numbers in general".  I don't see his proof (if he had or was aware of one), but I would imagine it be something along the lines that for every even number N there is an odd number N+1.  Therefore half of all numbers greater than 1 are odd.

I'm aware that zero is even, which makes me think there is one extra even number than odd numbers.  My thinking is that if half of all positive numbers are odd and half of all negative numbers are odd, than leaving out zero, half of all integers are odd.  But when you add in the only non-positive, non-negative number, which is even, you have an extra even number.  (The programmer in me wants to add the concepts of -0 and +0 for symmetry.  The rest of me thinks that's nuts!)

However, I think my imagined platonic proof still works as long as zero is definitely even and won't work if it's either odd or neither parity.

Are either of these proofs valid?

While they are both countable, there is a more applicable mechanism for measuring how many natural numbers have a certain property. If $A \subset \mathbb{N}$, let $$A_n = \text{number of elements in A less or equal to n}$$ then we can define density of $A$ as $$\operatorname{density}(A) = \lim_{n\to\infty}  \frac{A_n}{n}$$Then the odd numbers have density $\frac{1}{2}$.