AUndefined — you cannot add cardinals of different sizes
Bℵ₁, by the absorption rule: κ + λ = max(κ, λ) for infinite cardinals
Cℵ₂, since you move up one level in the cardinal hierarchy
D2ℵ₀, since you are doubling the smaller infinity
For infinite cardinals κ ≥ λ, cardinal addition satisfies κ + λ = max(κ, λ) = κ. Since ℵ₁ > ℵ₀, we have ℵ₀ + ℵ₁ = ℵ₁. Infinite cardinal addition absorbs the smaller cardinal completely — this follows (under AC) from the ability to exhibit an explicit bijection between ℵ₁ ⊔ ℵ₀ and ℵ₁ by interleaving the countably many extra elements into the already-uncountable ℵ₁.
Question 2 Multiple Choice
A student argues: 'ω + ω > ω as ordinals, so ℵ₀ + ℵ₀ > ℵ₀ as cardinals, since ω and ℵ₀ name the same set.' What is wrong with this reasoning?
ANothing — ordinal and cardinal arithmetic agree on the natural numbers
Bω and ℵ₀ name different sets; ω is finite and ℵ₀ is infinite
COrdinal and cardinal arithmetic are different operations: ω + ω compares well-ordered types while ℵ₀ + ℵ₀ asks only whether a bijection exists between ℕ ⊔ ℕ and ℕ
Dω + ω = ω in ordinal arithmetic, so the premise is false
ω and ℵ₀ name the same underlying set (ℕ) but represent different mathematical structures. ω is ℕ viewed as a well-ordered type, and ordinal addition is sensitive to order: ω + ω is a well-order where you count through one copy of ℕ then start again, producing a strictly longer well-order. Cardinal addition asks only whether a bijection exists between ℕ ⊔ ℕ and ℕ — which it does (map evens to one copy, odds to the other). Cardinality ignores order; ordinality preserves it.
Question 3 True / False
Since ℵ₀ + ℵ₀ = ℵ₀ and ℵ₀ · ℵ₀ = ℵ₀, it follows that 2^ℵ₀ = ℵ₀ as well.
TTrue
FFalse
Answer: False
Cardinal exponentiation behaves fundamentally differently from addition and multiplication. Cantor's theorem proves that for any set A, |P(A)| > |A| — the power set is always strictly larger. Since 2^ℵ₀ = |P(ℕ)|, we have 2^ℵ₀ > ℵ₀. The absorption rule that collapses addition and multiplication does NOT apply to exponentiation. This is the crucial asymmetry: infinite cardinal addition and multiplication are trivial, but exponentiation is genuinely nontrivial.
Question 4 True / False
The continuum hypothesis — that 2^ℵ₀ = ℵ₁ — was proven true by Kurt Gödel.
TTrue
FFalse
Answer: False
Gödel proved that the continuum hypothesis (CH) cannot be disproved from ZFC — it is consistent with ZFC. Later, Paul Cohen proved that CH also cannot be proved from ZFC. Together, these results show CH is independent of ZFC: neither a theorem nor a refutable claim within standard set theory. There are models of ZFC where 2^ℵ₀ = ℵ₁, and models where 2^ℵ₀ = ℵ₂₃₇. Gödel established consistency, not truth.
Question 5 Short Answer
Why does infinite cardinal addition collapse (κ + λ = max(κ, λ)) while cardinal exponentiation does not, and what does this reveal about the structure of infinite sets?
Think about your answer, then reveal below.
Model answer: Cardinal addition κ + λ asks whether we can biject κ ⊔ λ back to max(κ, λ). For infinite κ ≥ λ, we can — use AC to well-order κ, then interleave the λ elements into it. Infinity absorbs the addition. Exponentiation κ^λ counts functions from λ to κ. For 2^ℵ₀, this is the set of all binary sequences on ℕ — equivalently all subsets of ℕ. Cantor's diagonal argument shows no injection from P(ℕ) into ℕ exists, so the two cardinalities are genuinely different. Exponentiation generates new structure through combinatorial enumeration that cannot be bijected away.
This asymmetry is mathematically deep: it explains why the continuum hypothesis is non-trivial and why ZFC is 'incomplete' with respect to cardinal arithmetic. Infinite addition and multiplication are fully determined by ZFC; exponentiation opens a space of genuine independence where additional axioms would be needed to pin down the answer.