 Solution

ID: past-exam-of-the-analytic-topology-and-axiomatic-set-theory-course-of-the-university-of-oxford/2010/b/5/c/solution

ℵ_{0} = ω, ℵ_{α + 1} = (ℵ_{α})^{+}, ℵ_{λ} = α < λ sup ℵ_{α} (λ a nonzero limit ordinal) .

Here

κ^{+}

is the least cardinal strictly larger than

κ

2^{κ} = κ^{+} for every infinite cardinal κ,

equivalently

2^{ℵ_{α}} = ℵ_{α + 1}

for every ordinal

α

in ZFC.

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