Questions: Ordinal Arithmetic, Multiplication, and Exponentiation

Question 1

What is the ordinal 1 + ω?

• A. ω + 1, since adding to an infinite ordinal always produces something strictly larger
• B. ω, because placing one element before an infinite ascending sequence leaves the order type unchanged
• C. 2 × ω, because you are combining one copy of 1 with one copy of ω
• D. ω + 2, since each additional element on either side increases the ordinal by 1

Answer: (B)

Ordinal addition α + β means 'place a copy of α, then a copy of β after it.' So 1 + ω places one element, then appends ω after it. The result has a least element (the initial 1) followed by an infinite ascending chain — which still has order type ω. There is no new greatest element, and ω has no maximum. This contrasts with ω + 1 (place ω first, then append 1), which gives a greatest element and has order type strictly greater than ω. The key: it is not how many elements you add, but *where* relative to the existing structure.

Question 2

What is the ordinal 2 × ω?

• A. ω × 2 = ω + ω, since multiplication is commutative for infinite ordinals
• B. 4, since 2 × 2 = 4 and ω is just a large number
• C. ω, because ω many copies of 2 concatenated form a simple infinite ascending chain
• D. ω², since multiplying by ω produces the next level of the ordinal hierarchy

Answer: (C)

Ordinal multiplication α × β means 'β copies of α laid end-to-end.' So 2 × ω means ω many copies of 2 concatenated: {0,1 | 0,1 | 0,1 | ...}. Each pair is finite, and the whole sequence is still a simple ω-length chain — order type ω. Compare ω × 2, which means 2 copies of ω: the first ω followed by a second ω, giving order type ω + ω > ω. So 2 × ω = ω ≠ ω × 2. Left-multiplication 'repeats' the left argument in a way that collapses; right-multiplication 'scales' it, creating something genuinely larger.

Question 3

For infinite ordinals, ω + 1 = 1 + ω, since addition of infinite quantities is commutative.

• T. True
• F. False

Answer: False

Ordinal addition is non-commutative: 1 + ω = ω (adding one element before an infinite sequence leaves the order type as ω), while ω + 1 > ω (appending one element after ω creates a new greatest element, a strictly larger ordinal). The order — which copy comes first — determines the result because what matters is the order type, not just the cardinality. This non-commutativity is one of the sharpest ways ordinal arithmetic differs from natural number arithmetic.

Question 4

The ordinal ω^ω is an uncountable ordinal, since raising ω to the ω power produces something beyond countable infinity.

• T. True
• F. False

Answer: False

ω^ω is countable. It can be understood as the set of finite sequences of natural numbers ordered lexicographically — a countable set. The first uncountable ordinal is ω₁ (aleph-one), which is far beyond ω^ω in the ordinal hierarchy. Countability is a cardinality concept, not an ordinal concept: ω^ω is much 'larger' than ω as an ordinal (as an order type), but the underlying set is still countable. The famous ordinal ε₀ = ω^(ω^(ω^⋯)) is also still countable.

Question 5

Explain why ordinal addition is non-commutative, using 1 + ω versus ω + 1 as your example. What fundamental property of ordinals makes these two expressions different?

Model answer: Ordinal addition is defined by order type — the arrangement of elements, not just their count. α + β means 'place a copy of α, then a copy of β.' In 1 + ω, one element precedes ω; the result has a least element but no greatest element and order type ω. In ω + 1, ω precedes one element; the result has a new greatest element (no element in ω is greatest), giving order type ω + 1 > ω. The same elements are present in both cases, but their arrangement differs — and ordinals track arrangement, not cardinality.

This non-commutativity is the key insight of ordinal arithmetic. Ordinals are defined as order types, so two sets with the same elements but different orderings can be different ordinals. Adding 1 'before' ω slots it into a position that already has something infinite coming after it — no new structure is created. Adding 1 'after' ω extends the sequence beyond its previous supremum — genuinely new structure appears. In ordinary arithmetic, 1 + n = n + 1 because we only count; in ordinal arithmetic, we track where.