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[The other answers have more than answered my question, so my goal in self-answering is to make sure I understand what's been said so far. I would appreciate corrective comments if I make any mathematical mistakes. I'm heavily indebted to JDH's excellent answer to the question "Is infinity an odd or even number?"]

For numbers greater than one, which is the set Socrates refers to, even numbers are those that can be divided on the two sides of a scale so that the scale balances. If I have 10 identical coins (an even number), I can put 5 on each side. But if I add (or subtract) one coin from the total, the scale will not balance. (Something is "odd" about the number.)

Moving up a level of abstraction, we notice that an even number of any identical object will balance: 10 coins, 10 jars of olive oil, 10 grains of sand, etc. So depending on your point of view, evenness is a property of 10 or 10 is an instance of Even. Even when we consider things that we can't reasonably weigh on a scale, such as imaginary objects, can still be labeled as even if there are an even number of them.

So what happens if you take numbers and metaphorically divide them between the two pans of a scale: even numbers in one and odd numbers in the other? It depends on the range of numbers you consider. [2...3] will balance, but [2...4] won't. But you can always add one more number to the range and make the scale balance again, so [2...5] does balance.

Now Socrates didn't have a concept of infinity, so he probably was thinking of arbitrarily large numbers. If you happen to pick an odd number of numbers, than one or the other pan on the hypothetical scale will be tip. But since the range is arbitrary, you can always pick just one more number and restore the balance. Nothing about zero or infinity or negative numbers changes this: if you want half of all numbers to be odd, just make sure your range includes an even number of numbers.

When you start talking about infinity or an infinite number of numbers and so on, things get weird and you can't really apply everyday intuition. There are any number of paradoxical-seeming results that you can arrive at if you start messing around with infinite series. For instance, you can have a hotel with an infinite number of rooms, an infinite number of guests, and plenty of room for an infinite number of new guests. More to the point, you can match an infinite number of odd numbers to an infinite number of even numbers at any ratio you like without running out of either type of number. So saying that half of all numbers is odd is just as true as saying that 1 in a thousand numbers are odd.

However, we can still look back at our scale and for large ranges of numbers, the scale either balances or very-nearly balances for any particular range you pick. Sure there is one extra even number in the range [2...1,000] , but it would take a pretty accurate scale to determine which pan is weighed down more. And the larger the range, the smaller the difference. Mathematically, this way of thinking corresponds to the concept of natural density. Thankfully, the natural density of the odd numbers turns out to be half when measured this way.

As an aside, Wikipedia mentions that:

This notion can be understood as a kind of probability of choosing a number from the set A. Indeed, the asymptotic density (as well as some other types of densities) is studied in probabilistic number theory.

Summary

As long as we steer away from the (very intriguing) notion of infinity, it seems that Socrates was right to say that half of the numbers are odd.