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Full Idea
The axiom of choice is essential for proving the downward Löwenheim-Skolem Theorem.
Gist of Idea
Choice is essential for proving downward Löwenheim-Skolem
Source
Stewart Shapiro (Foundations without Foundationalism [1991], 4.1)
Book Ref
Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.80