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human
Are there more primes than non-primes?

There is an infinite number of integers divisible by 3, and for each of them there are two integers not evenly divisible by 3. Given that both are infinite, can you say that there are "more" numbers not divisible by 3? 
And further, if you can say that, can it be shown whether or not there are more primes than non-primes? 

Thanks in advance.
gpt
Since both sets are infinite, it depends on what you mean by "more". One common sense of size mathematicians use to describe sets is called "cardinality". Two sets have the same cardinality if you can define a function from one to the other such that each element in one set corresponds to exactly one element in the other. The primes and non-primes are both "countably infinite", meaning they have the same cardinality as the integers, and so they are the same size as each other. In this sense, there are the same number of primes as non-primes. In the same sense, there are also the same amount of number divisible by 3 as numbers not divisible by 3.

Another sense of "size", which only applies sets of (positive) integers, is to define a_n as the number of (positive) integers m in the set S that are less than n, then take the limit of a_n / n as n goes to infinity. This is sometimes called the "asymptotic density" of the set. The asymptotic density of the numbers divisible by 3 is 1/3, and the asymptotic density of the numbers not divisible by 3 is 2/3, so in this sense more numbers are not divisible by 3 than are. For primes, a_n ~ n / log(n), so the limit of a_n / n is 0. Naturally, non-primes are the complement of the primes, and so the asymptotic density of the non-primes is 1. Thus, in this sense, there are actually many more non-primes than primes.

It should not be surprising that there are more non-primes than primes. Even numbers and odd numbers both have asymptotic densities of 1/2, and only one even number is prime, while many odd numbers are non-prime. If you consider numbers divisible by neither 2 nor 3, they have an asymptotic density of only 1/3. This is already less than half the integers, and will only get smaller when we include more prime factors.