 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-24/6/a/i/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

M

when every ground-model cardinal remains a cardinal in every generic extension:

\forall G generic over M \forall κ \in M M ⊨ “ κ is a cardinal” \Rightarrow M [G] ⊨ “ κ is a cardinal” .

Forcing preserves ordinals, and an ordinal that was not a cardinal already has a ground-model bijection with a smaller ordinal, which persists. Thus this definition gives equality of the cardinal ordinals in the two models.

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