Questions: Cardinal Arithmetic for Infinite Sets

Question 1

Which of the following correctly describes ℵ₀ × ℵ₀?

• A. ℵ₁ — multiplying infinities always produces a strictly larger infinity
• B. ℵ₀ — the set of all pairs of natural numbers is still countably infinite
• C. 2^ℵ₀ — the product of a set with itself equals its power set
• D. Undefined — multiplication of infinite cardinals is not a valid operation

Answer: (B)

ℵ₀ × ℵ₀ = ℵ₀. The product ℕ × ℕ (all pairs of natural numbers) can be put in bijection with ℕ using Cantor's diagonal enumeration: list pairs (0,0), (1,0), (0,1), (2,0), (1,1), (0,2), … This shows the infinite grid of pairs is countable. In general, κ × κ = κ for any infinite cardinal. The intuition from finite arithmetic — that a grid is strictly larger than a line — breaks down completely for infinite sets.

Question 2

What does Cantor's theorem guarantee about 2^ℵ₀, and what remains undecidable from ZFC alone?

• A. Cantor's theorem proves 2^ℵ₀ = ℵ₁; ZFC fully determines the exact value
• B. Cantor's theorem proves 2^ℵ₀ > ℵ₀; which specific aleph 2^ℵ₀ equals cannot be proved or disproved from ZFC
• C. Cantor's theorem proves 2^ℵ₀ = ℵ₀; the apparent size difference is an artifact of diagonal arguments
• D. Cantor's theorem only applies to finite sets; for infinite sets 2^ℵ₀ is undefined

Answer: (B)

Cantor's theorem states that for any set A, |P(A)| > |A| — the power set is strictly larger. Applied to ℕ, this gives 2^ℵ₀ > ℵ₀, proved via the diagonal argument. But ZFC cannot decide exactly which aleph 2^ℵ₀ equals. The Continuum Hypothesis (CH: 2^ℵ₀ = ℵ₁) is independent of ZFC — Gödel showed CH is consistent with ZFC (1940) and Cohen showed its negation is also consistent (1963). This independence result is one of the deepest in 20th-century mathematics.

Question 3

2^ℵ₀ is strictly greater than ℵ₀.

• T. True
• F. False

Answer: True

This is Cantor's diagonal theorem. Suppose there were a bijection f: ℕ → P(ℕ). Construct D = {n ∈ ℕ : n ∉ f(n)}. D is a subset of ℕ, so D ∈ P(ℕ), but D differs from every f(n) at position n — no preimage exists, contradicting bijectivity. Therefore |P(ℕ)| = 2^ℵ₀ > ℵ₀. This result holds for every cardinal κ: 2^κ > κ always, making iterated exponentiation the ladder that produces strictly larger infinities.

Question 4

For infinite cardinals, cardinal addition is just as complex and undecidable as cardinal exponentiation.

• T. True
• F. False

Answer: False

Cardinal addition for infinite sets is completely trivial: if κ is infinite and λ ≤ κ, then κ + λ = κ. In particular ℵ₀ + ℵ₀ = ℵ₀, and κ + κ = κ for any infinite κ. The same collapse occurs for multiplication: κ × κ = κ. Cardinal exponentiation, by contrast, is genuinely complex: which aleph 2^κ equals is undecidable from ZFC alone. The central asymmetry of infinite cardinal arithmetic is that addition and multiplication are boring (fully determined, always equal to the larger operand), while exponentiation is wild and partially undecidable.

Question 5

Explain why cardinal addition and multiplication are 'trivial' for infinite sets while cardinal exponentiation is not, and what this asymmetry reveals about the structure of infinite cardinals.

Model answer: For infinite cardinals, κ + λ = κ × λ = max(κ, λ) — you can always biject the union or Cartesian product of two infinite sets of the same size back onto one factor. Infinite sets are 'large enough' to absorb copies of themselves without growing. Exponentiation is different: 2^κ counts all subsets of a κ-sized set (the power set), which Cantor proved is always strictly larger than κ by the diagonal argument. Moreover, exactly how much larger depends on additional axioms (like the Continuum Hypothesis) that are independent of ZFC. The asymmetry reveals that the hierarchy of infinite sizes is driven primarily by iterated power sets, not by addition or multiplication.

Students who only memorize 'ℵ₀ + ℵ₀ = ℵ₀' miss the deeper point: the triviality of addition and multiplication is precisely what makes exponentiation stand out as genuinely interesting and genuinely hard. The undecidability of the Continuum Hypothesis is the most dramatic consequence of this asymmetry — the most basic question about 2^ℵ₀ cannot be answered within standard set theory.