Sentence examples similar to a cardinal hypothesis from inspiring English sources
The so-called Singular Cardinal Hypothesis (SCH) completely determines the behavior of the exponentiation for singular cardinals, modulo the exponentiation for regular cardinals.
Modulo still-open conjectures on the existence of iteration strategies, this lemma yields models M of the Axiom of Determinacy such that HOD^M can be analyzed fine structurally (for example, satisfies the GCH), and yet satisfies very strong large cardinal hypotheses (for example, that there are superstrong cardinals).
The interesting thing about these large cardinal hypotheses is that for n > 0 they are inconsistent with AC (by a result of Kunen).
Nowadays, large cardinal hypotheses are really taken to mean some kind of embedding properties between the set theoretic universe and inner models of set theory.
For suppose that \(\mathsf{T}\) is a recursively axiomatic theory which is sufficiently strong to formalize our current mathematical theorizing – e.g. Zermelo Fraenkel set theory with the Axiom of Choice [\(\mathsf{ZFC}\)], supplemented as needed with large cardinal hypotheses.
He is an archbishop, not a cardinal.
Now a cardinal was speaking very differently.
He was made a cardinal in 1983.
(He became a cardinal in 1983).
