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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).
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.
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.
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