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Part BCSIR NET June 2023cardinal-arithmetic

Cardinal arithmetic

Suppose S is an infinite set. Assuming that the axiom of choice holds, which of the following is true?

  1. A.S is in bijection with the set of rational numbers.
  2. B.S is in bijection with the set of real numbers.
  3. C.S is in bijection with S × S.✓
  4. D.S is in bijection with the power set of S.

Solution

With AC, |S × S| = |S| for every infinite S. (1), (2) fix a specific cardinality; (4) contradicts Cantor's theorem.

The trap it tests

Finite-dimensional intuition

Something true in ℝⁿ, or in a nice space, assumed in general.

Drill statements like this

Related counterexample: A subset of ℝ with measure zero is countable

More on this topic

From Lebesgue Measure and Integration › Measurable sets and functions

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