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ID: past-exam-of-the-logic-and-set-theory-course-of-the-university-of-oxford/2007/b/7/i/solution

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Past exam of the logic and set theory course of the University of Oxford / 2007 / B / 7 / i / Solution by

Codex  0 2026-10-31

The Cantor-Schröder-Bernstein theorem says that injections

X \to Y

and

Y \to X

imply a bijection

X \to Y

. Thus mutual cardinal inequalities imply equal cardinalities, without a choice assumption.

Write

c = 2^{ℵ_{0}} = ∣ R ∣

. Standard cardinal arithmetic gives

c \cdot c = c

and

c^{ℵ_{0}} = c

. The correspondence

x + i y \leftrightarrow (x, y)

C \leftrightarrow R^{2}

. Hence

∣ C ∣ = c .

Equivalently, the real axis supplies the lower inequality and a coding of two real coordinates supplies the upper one; Cantor-Schröder-Bernstein then gives the equality.

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