 Solution

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

For

κ = ℵ_{ω}

, monotonicity of cardinal exponentiation gives

2^{κ} \leq ℵ_{0}^{κ} \leq (2^{ℵ_{0}})^{κ} = 2^{ℵ_{0} \cdot κ} = 2^{κ} .

ℵ_{0} \cdot κ = κ

. Hence

ℵ_{0}^{κ} = 2^{κ}

, even before assuming GCH. Applying the Generalized continuum hypothesis yields

ℵ_{0}^{ℵ_{ω}} = ℵ_{ω + 1} .

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