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We understand that sets have a cardinality , that is, that collections have a number associated with them and it doesn't really matter what the members of that set are.
Hausdorff proved further results on the cardinality of Borel sets in 1916.
Frege's approach to providing a logical analysis of cardinality , the natural numbers, infinity and mathematical induction were groundbreaking, and have had a lasting importance within mathematical logic.
The Skolem-Lowenheim theorem asserts that any first-order theory having an infinite model has other models of all smaller infinite cardinalities .
Cantor conjectured that there are no infinite cardinalities between the size of the natural numbers and the size of the real numbers (and so there are no sets S as described above).
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