Questions: Infinite Cardinal Numbers

Question 1

Which of the following correctly describes the relationship between ℵ₁ and the cardinality of the real numbers |ℝ|?

• A. ℵ₁ = |ℝ| by definition, since ℝ is the first uncountable set constructed in standard mathematics
• B. ℵ₁ < |ℝ| always, because 2^ℵ₀ is strictly larger than the first uncountable cardinal
• C. Whether ℵ₁ = |ℝ| is independent of ZFC — it can neither be proved nor disproved from the standard axioms
• D. ℵ₁ > |ℝ| because ω₁ contains more elements than the continuum

Answer: (C)

By Cantor's theorem, |ℝ| = 2^ℵ₀ > ℵ₀, so ℝ is uncountable. But ℵ₁ is defined as the *second* infinite cardinal — the first uncountable one — purely from ordinal theory, with no reference to ℝ. Whether 2^ℵ₀ = ℵ₁ (the continuum hypothesis) is a statement that Gödel showed cannot be disproved from ZFC and Cohen showed cannot be proved from ZFC. It is independent — both CH and ¬CH are consistent with ZFC.

Question 2

ℵ₁ is by definition the cardinality of the real numbers ℝ.

• T. True
• F. False

Answer: False

ℵ₁ is defined as the first uncountable cardinal — the unique initial ordinal ω₁, the smallest ordinal with no bijection to ω. The cardinality of ℝ is 2^ℵ₀, which is provably uncountable (by Cantor's diagonal argument) and thus at least ℵ₁. Whether |ℝ| = ℵ₁ exactly is the continuum hypothesis, which is independent of ZFC. The definition of ℵ₁ makes no reference to ℝ.

Question 3

Explain the difference between a cardinal number and an ordinal number, and why every cardinal is an ordinal but not conversely.

Model answer: Ordinals measure order-type: they represent the isomorphism classes of well-ordered sets. Cardinals measure cardinality: two sets have the same cardinal if and only if there is a bijection between them. Every cardinal is an initial ordinal — an ordinal α such that no β < α has a bijection with α. But many ordinals are not initial: ω and ω+1 are different ordinals with the same cardinality ℵ₀, so ω+1 is not a cardinal. The infinite cardinals ℵ₀, ℵ₁, ℵ₂, ... form a proper subclass of the ordinals.

The key distinction is between order structure (which ordinals track) and mere size (which cardinals track). The ordinals ω, ω+1, ω+2, ... ω·2, ... are all countably infinite and thus have the same cardinality ℵ₀ even though they are all distinct ordinals. The cardinal ℵ₀ = ω is the first infinite initial ordinal; ℵ₁ = ω₁ is the next.