Aω + 1, because addition is commutative for infinite ordinals
Bω, because placing one element before an infinite sequence leaves the order type unchanged
Cω · 2, because adding a finite ordinal to an infinite one doubles the structure
D2, because the single element and the first element of ω merge into one
1 + ω means: one element, followed immediately by ω (the natural numbers in order). The result is: one element, then 0, 1, 2, 3, ... This sequence has no last element, and every element (except the initial one) has a predecessor — it is order-isomorphic to ω itself. The initial element gets 'absorbed' into the beginning of the infinite sequence. In contrast, ω + 1 places ω first and then appends a final element, creating an order type with a greatest element — genuinely different from ω. This asymmetry is why 1 + ω = ω but ω + 1 ≠ ω.
Question 2 Multiple Choice
A student claims ω · 2 = 2 · ω because 'multiplication of infinite sets should be commutative — both are just countably infinite.' What is wrong with this reasoning?
ANothing is wrong — ω · 2 and 2 · ω are equal as ordinals, though the reasoning is imprecise
BOrdinal multiplication is defined for finite ordinals only; infinite cases require cardinal arithmetic
COrdinals represent order types, not just sizes — ω · 2 (two copies of ω in sequence) and 2 · ω (ω copies of the pair {0,1}) have different structures, even though both are countable
DThe student is correct that they are equal in size but wrong that this makes them equal as ordinals — ordinals are always unequal unless they are finite
The student's error is conflating size (cardinality) with structure (order type). ω · 2 = ω + ω: two infinite sequences in tandem, with a specific 'gap' between them — the second copy begins after all elements of the first. 2 · ω = ω copies of {0, 1}, producing an endless sequence of pairs — order-isomorphic to ω itself. These are different order types. The cardinal ℵ₀ · 2 = ℵ₀ is true, but ordinal arithmetic is strictly more fine-grained than cardinal arithmetic. The same-size reasoning works for cardinals, not ordinals.
Question 3 True / False
Ordinal addition is not commutative: for some ordinals α and β, α + β ≠ β + α.
TTrue
FFalse
Answer: True
The canonical example is 1 + ω ≠ ω + 1. Ordinal addition α + β means: place a copy of α, then immediately after it place a copy of β. Swapping the order changes which order type comes first in the concatenation, and this can change the resulting structure. 1 + ω has no greatest element (the initial element is followed by the endless ω), giving order type ω. But ω + 1 has a greatest element (the appended final element), giving a strictly larger order type. Non-commutativity is a fundamental feature of ordinal arithmetic, not an anomaly.
Question 4 True / False
Since ω and ω + 1 are both countably infinite sets, they represent the same ordinal.
TTrue
FFalse
Answer: False
This is the core misconception the topic addresses. Ordinals are not just measures of size — they encode order type, the structural pattern of how elements are arranged. ω is the order type of a set with no greatest element where every element has finitely many predecessors. ω + 1 is the order type of a set with a greatest element (and all other properties of ω). These are not order-isomorphic — no bijection between them preserves order — so they are distinct ordinals, even though both are countably infinite. Cardinals and ordinals diverge exactly at this point.
Question 5 Short Answer
Why does 1 + ω = ω, while ω + 1 ≠ ω? Explain using the concept of order types.
Think about your answer, then reveal below.
Model answer: Ordinal addition α + β means: concatenate an order-isomorphic copy of α with a copy of β, in that sequence. For 1 + ω: one element followed by the natural numbers. The resulting order has no greatest element and every element has finitely many predecessors — it is structurally identical to ω. For ω + 1: the natural numbers followed by one final element. The resulting order has a greatest element, which ω does not. Since ω and ω + 1 are not order-isomorphic (no bijection preserves order between them), they are different ordinals. The order of concatenation determines the structure of the result.
The key is that 'adding' in ordinal arithmetic means concatenating sequences, not counting items. When you add a finite beginning to an infinite sequence, the infinite sequence overwhelms the finite part — the result is still just ω. But when you append to the end of an infinite sequence, you create a new last element that did not exist before, which is a genuine structural change. This is why order matters: what comes first can be swallowed by what follows, but what comes last always leaves a trace.