I have no problem with the idea that mathematicians can improve on definitions and make them more natural, especially as they learn more abstraction.
And I don't think it was after the fact, or a generalisation of anything, to say that the natural numbers are the classes of sets with their own cardinality. It was the definition given by Frege, who I believe was the first person with the audacity to define something so primitive. And defining natural numbers in terms of their cardinality strikes me as entirely natural, once you have it that natural numbers are just the finite numbers you use to count stuff (i.e. cardinal numbers --- counting stuff by forming one-one correspondences).
Frege's definition is not the one given by mathgrad, because it doesn't work in ZFC. We can't use equivalence classes, so instead, we pick out canonical representatives of the classes, which is a good enough compromise.
That said, I don't think there is one definition of natural numbers to rule them all. I am personally quite fond of the Church numerals, where a natural number n is defined as the higher-order function which composes its argument with itself n times. I prefer the thought that what counting is about is the repetition of a single operation.
As for the rest, I am the spineless type of formalist who is happy to throw his hands up in the final analysis and say that all of mathematics is arbitrary. And so unlike the OP, I won't draw a distinction between 0^0=1, 1+1=2 or 0.999...=1. You can mangle all of these as much as you like, but so long as you keep the expressive power of arithmetic, you'll find an analogue of Turing equivalency to translate to the original.
And I don't think it was after the fact, or a generalisation of anything, to say that the natural numbers are the classes of sets with their own cardinality. It was the definition given by Frege, who I believe was the first person with the audacity to define something so primitive. And defining natural numbers in terms of their cardinality strikes me as entirely natural, once you have it that natural numbers are just the finite numbers you use to count stuff (i.e. cardinal numbers --- counting stuff by forming one-one correspondences).
Frege's definition is not the one given by mathgrad, because it doesn't work in ZFC. We can't use equivalence classes, so instead, we pick out canonical representatives of the classes, which is a good enough compromise.
That said, I don't think there is one definition of natural numbers to rule them all. I am personally quite fond of the Church numerals, where a natural number n is defined as the higher-order function which composes its argument with itself n times. I prefer the thought that what counting is about is the repetition of a single operation.
As for the rest, I am the spineless type of formalist who is happy to throw his hands up in the final analysis and say that all of mathematics is arbitrary. And so unlike the OP, I won't draw a distinction between 0^0=1, 1+1=2 or 0.999...=1. You can mangle all of these as much as you like, but so long as you keep the expressive power of arithmetic, you'll find an analogue of Turing equivalency to translate to the original.