Aω + 1 — ordinal addition is commutative, just as in natural number arithmetic
Bω — the single element placed before the infinite sequence is absorbed and the order type is unchanged
CA new ordinal strictly between ω and ω + 1
DUndefined — you cannot add a finite ordinal and a transfinite ordinal
Ordinal addition is defined by concatenation: 1 + ω means place one element first, then append the sequence {0,1,2,...}. The result is {*, 0, 1, 2,...} — an infinite sequence with one extra element at the very start. Every element still has only finitely many predecessors; the order type is ω. Compare this to ω + 1, where one element is placed after all of ω: that new element has infinitely many predecessors, creating an order type strictly greater than ω. The finite element 'disappears' when prepended to a limit ordinal.
Question 2 Multiple Choice
Why is 1 + ω = ω but ω + 1 ≠ ω in ordinal arithmetic?
ABecause ω is the smallest infinite ordinal, and adding anything smaller than ω to it cannot produce a larger ordinal
BBecause ordinal addition is defined by concatenating well-orderings: prepending one element to ω leaves the order type as ω, but appending one element after ω creates a last element with infinitely many predecessors — a strictly new ordinal
CBecause ordinal arithmetic always reduces to the larger operand when the smaller operand is finite
DBecause 1 < ω, and in ordinal arithmetic the smaller summand is always absorbed by the larger one
The asymmetry follows directly from the definition. In 1 + ω, the lone element sits at position 0 in the concatenated sequence, and every subsequent element has a finite number of predecessors — indistinguishable from ω itself. In ω + 1, the appended element comes after all of ω: it has no immediate predecessor among the naturals, has infinitely many predecessors, and is not reachable by any finite number of steps from 0. Option D is tempting but wrong: the result is not always the larger operand — it depends on which side the finite ordinal appears on. 2 + ω = ω but ω + 2 = ω + 2.
Question 3 True / False
In ordinal arithmetic, ω · 2 = ω + ω, which is strictly larger than ω.
TTrue
FFalse
Answer: True
Ordinal multiplication α · β means 'replace each element of β with a copy of α.' So ω · 2 replaces each of the two elements of 2 with a copy of ω, producing a well-ordering of type ω + ω: {0,1,2,...} followed by {0',1',2',...}. This has a point (0') that has infinitely many predecessors (all of the first copy), making it a strictly larger ordinal than ω. It is also larger than ω + 1, ω + 2, ..., sitting above all ω + n for finite n.
Question 4 True / False
Since ℵ₀ + ℵ₀ = ℵ₀ in cardinal arithmetic, it follows that ω + ω = ω in ordinal arithmetic.
TTrue
FFalse
Answer: False
Ordinal arithmetic and cardinal arithmetic are fundamentally different. Cardinals measure set size (cardinality); ordinals measure order type (the structure of the well-ordering). The sets underlying ω and ω + ω have the same cardinality ℵ₀ — both are countably infinite — so as cardinals they are equal. But as ordinals they are distinct: ω + ω has a point (the start of the second copy) with infinitely many predecessors, which ω does not. Ordinal arithmetic preserves order structure; cardinal arithmetic collapses it. The fact that ordinals can differ while having the same cardinality is precisely why both concepts are needed.
Question 5 Short Answer
Explain in your own words why ordinal addition is not commutative, using the definition of ordinal addition as concatenation of well-orderings.
Think about your answer, then reveal below.
Model answer: Ordinal addition α + β means 'place a copy of α first, then append a copy of β.' Since well-orderings have a fixed direction, the order of operands determines which elements come first. When computing 1 + ω: one element is placed at the start of the infinite sequence. The result has the same order type as ω because the lone starting element still has only finitely many predecessors — it's just the 'first natural number.' When computing ω + 1: one element is appended after all of ω. This new element has infinitely many predecessors (all of ω), which is something ω itself doesn't contain. The two concatenations produce different order types, so the operation is non-commutative.
The deeper principle is that ordinals are not just numbers — they encode order structure. Adding on the left can get 'swallowed' by a sufficiently large limit ordinal, while adding on the right always creates a new distinct position. This is why non-commutativity is a structural fact about the definition of ordinal addition, not an accident. The same non-commutativity extends to ordinal multiplication and exponentiation.