> I believe that it’s defined as the size of ω₁, the set of wellorderings of the natural numbers. I don’t think there’s any real sense in which that’s less of a concrete set than the power set of the natural numbers, even if it’s harder to picture.
The definition with which I am familiar is that \aleph_1 is the least cardinal that is strictly greater than \aleph_0, but, of course, one can use any of an equivalent collection of properties as the definition.
Nonetheless, at least any definition in line with usual mathematical practice should probably not provably require that \aleph_1 is at least 2^{\aleph_0}, which I claim that your definition does. Let me refer to positive integers, and non-negative integers, to avoid having to commit to what the natural numbers are. I define a function from the power set of the positive integers to the set of total orderings of the non-negative integers as follows. For every power set A, write B for the complement of A in the positive integers, and let <_A be the order that restricts to the usual order on A and on B (though not on their union), and that declares 0 to be greater than every element of A and less than every element of B. Since the usual order on the positive integers is a well order, so is <_A. Clearly, A may be recovered from <_A (as the set of elements preceding 0), so the function is injective.
Sorry, classes of well-orderings of subsets of the natural numbers up to order isomorphism: https://en.wikipedia.org/wiki/Hartogs_number. And yes it’s equivalent to your definition due to ordinal magic as far as I can tell (or maybe because cardinals are by definition well ordered, I’m clearly not an expert).
The definition with which I am familiar is that \aleph_1 is the least cardinal that is strictly greater than \aleph_0, but, of course, one can use any of an equivalent collection of properties as the definition.
Nonetheless, at least any definition in line with usual mathematical practice should probably not provably require that \aleph_1 is at least 2^{\aleph_0}, which I claim that your definition does. Let me refer to positive integers, and non-negative integers, to avoid having to commit to what the natural numbers are. I define a function from the power set of the positive integers to the set of total orderings of the non-negative integers as follows. For every power set A, write B for the complement of A in the positive integers, and let <_A be the order that restricts to the usual order on A and on B (though not on their union), and that declares 0 to be greater than every element of A and less than every element of B. Since the usual order on the positive integers is a well order, so is <_A. Clearly, A may be recovered from <_A (as the set of elements preceding 0), so the function is injective.