Where do infinite sets come from?

I only have a rudimentary understanding of sets, here's what I know: We can specify a set by naming its elements, for example A={1,2,3} We can specify a set by naming a set X and asking which elements x in X satisfy P(x). For example B={x in A| x is even}. But from here how do we get to sets with... show more I only have a rudimentary understanding of sets, here's what I know:

We can specify a set by naming its elements, for example A={1,2,3}
We can specify a set by naming a set X and asking which elements x in X satisfy P(x). For example B={x in A| x is even}.

But from here how do we get to sets with infinite size, like the natural numbers or the reals?
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