Questions: Cardinal Arithmetic, Exponentiation, and Hierarchy

Question 1

For an infinite cardinal κ, which of the following correctly simplifies κ + κ?

• A. 2κ — the result is twice the original cardinality, as in finite arithmetic
• B. κ — addition collapses for infinite cardinals; the sum is no larger than either summand
• C. κ² — addition behaves like multiplication for infinite cardinals
• D. It depends on which infinite cardinal κ is — different infinite cardinals behave differently under addition

Answer: (B)

Cardinal addition collapses for infinite cardinals: κ + κ = κ for any infinite κ. This is provable by bijection — for ℕ, you can interleave two copies of ℕ (send (n, 0) → 2n and (n, 1) → 2n+1) to show |ℕ ∪ ℕ| = |ℕ|. The same argument scales to any infinite cardinal. Finite intuition — where 2 + 2 = 4, not 2 — completely breaks down at infinity. Option D is wrong: the collapse holds for ALL infinite cardinals, not just some.

Question 2

A student claims: 'The set of all functions from ℕ to {0, 1} has the same cardinality as ℕ, because ℕ is infinite and infinite sets absorb additions.' What is wrong with this reasoning?

• A. Nothing — the claim is correct; all infinite sets are equinumerous with ℕ
• B. The flaw is that cardinal multiplication, not addition, is the relevant operation here
• C. The collapse law applies to addition and multiplication but not to exponentiation — the set of functions from ℕ to {0,1} has cardinality 2^ℵ₀, which is strictly greater than ℵ₀ by Cantor's theorem
• D. The claim would be correct if we used ordinal arithmetic instead of cardinal arithmetic

Answer: (C)

The student correctly notes that infinite cardinals absorb addition and multiplication. But the set of all functions from ℕ to {0,1} has cardinality 2^ℵ₀ — this is cardinal *exponentiation*, which does NOT collapse. Cantor's theorem guarantees |P(κ)| > |κ| for any set, and the set of all {0,1}-valued functions on ℕ bijects with P(ℕ). So 2^ℵ₀ > ℵ₀ strictly. Exponentiation is the one cardinal arithmetic operation that always produces something genuinely larger.

Question 3

For any infinite cardinal κ, the product κ × κ is strictly larger than κ.

• T. True
• F. False

Answer: False

Multiplication also collapses for infinite cardinals: κ × κ = κ for any infinite κ. The proof uses the Cantor pairing function, which is a bijection from ℕ × ℕ to ℕ, and generalizes via transfinite induction to all infinite cardinals. This means that the Cartesian product of an infinite set with itself has the same cardinality as the original set — a fact that has no analog in finite arithmetic (where 3 × 3 = 9 ≠ 3). Both addition and multiplication collapse; only exponentiation escapes this collapse.

Question 4

Cantor's theorem guarantees that for any cardinal κ, the cardinality of the power set P(κ) is strictly greater than κ, so 2^κ > κ for all infinite cardinals.

• T. True
• F. False

Answer: True

Cantor's theorem is the foundational result here. It states that no set can be put in bijection with its power set — |P(A)| > |A| for any set A, including infinite ones. Since 2^κ is defined as |P(κ)| (or equivalently, the cardinality of all functions from a set of size κ to {0,1}, which bijects with the power set), we get 2^κ > κ strictly. This is what makes exponentiation the engine of infinity-generation: it always produces a new, strictly larger infinite cardinal.

Question 5

Why does cardinal exponentiation produce genuinely new, larger infinities while cardinal addition and multiplication do not?

Model answer: Cardinal addition and multiplication collapse because infinite sets can absorb finite-like combinations of themselves: interleaving two copies of ℕ still gives a set the same size as ℕ (bijection via alternation), and the grid ℕ×ℕ can be put in bijection with ℕ via the Cantor pairing function. These bijections work because the combinatorial structure of addition and multiplication does not require the output to be 'bigger.' Exponentiation is different: 2^κ counts the number of distinct functions from a κ-sized domain to {0,1}, or equivalently all subsets of a κ-sized set. Cantor's diagonal argument proves no such bijection can exist — assuming one does and then constructing a function that differs from every listed function on at least one input produces a contradiction. The power set is irreducibly larger than the original set. Addition and multiplication stay within the same infinity; exponentiation always escapes to a strictly larger one.

This is why the beth hierarchy (ℶ₀ = ℵ₀, ℶ₁ = 2^ℵ₀, ℶ₂ = 2^(2^ℵ₀), ...) grows strictly with each step — each beth is the power set of the previous — while successor cardinals in the aleph hierarchy (ℵ₀, ℵ₁, ℵ₂, ...) may or may not coincide with beths, a question that ZFC cannot resolve (the Continuum Hypothesis).