I understand what you're saying, so thank you, but I still find myself disagreeing.
There are just as many odd numbers as even, so there's nothing more natural about either. They alternate. Yes you can extend to higher multiples, but there's still nothing more natural about multiples of 7 vs. multiples of 7 with remainder 3.
And it's just as easy to say that infinity is divisible by 7, as it is to say that infinity is divisible by 7 with remainder 3:
[1, 2, 3, 4, 5, 6, 7], [8, 9, 10, 11, 12, 13, 14], ...
1, 2, 3, [4, 5, 6, 7, 8, 9, 10], [11, 12, 13, 14, 15 16, 17], ...
So the entire idea I'm arguing against is that there's anything more natural, more default, more basic about the concept of "evenness" next to "oddness". The very first natural number, 1, is odd -- not even -- so it's just as easy to say that oddness comes first. But really they're fundamentally
complementary -- they require each other, neither is more primitive.
There are just as many odd numbers as even, so there's nothing more natural about either. They alternate. Yes you can extend to higher multiples, but there's still nothing more natural about multiples of 7 vs. multiples of 7 with remainder 3.
And it's just as easy to say that infinity is divisible by 7, as it is to say that infinity is divisible by 7 with remainder 3:
So the entire idea I'm arguing against is that there's anything more natural, more default, more basic about the concept of "evenness" next to "oddness". The very first natural number, 1, is odd -- not even -- so it's just as easy to say that oddness comes first. But really they're fundamentally complementary -- they require each other, neither is more primitive.