ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-29

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

$2 \cdot ω = ω$ while $ω \cdot 2 = ω + ω$ , pictured as order types

Example

$2 \cdot ω = ω, ω \cdot 2 = ω + ω > ω .$

Under the convention of Ordinal multiplication $α \cdot β$ , $α \cdot β$ is $β$ copies of $α$ ( $α \cdot β$ is the order type of $α \times β$ ordered by last differences, that is $β$ copies of $α$ ). So $2 \cdot ω$ is $ω$ copies of a two element set, laid end to end:

$\underset{}{\underset{⏟}{∙ ∙}} \underset{}{\underset{⏟}{∙ ∙}} \underset{}{\underset{⏟}{∙ ∙}} \dots$

which is a copy of $ω$ once the points are counted off $0, 1, 2, \dots$ . And $ω \cdot 2$ is two copies of $ω$ , one entirely above the other:

$0, 1, 2, \dots then 0^{'}, 1^{'}, 2^{'}, \dots$

which is $ω + ω$ and is strictly larger than $ω$ .

Both values are computed below from the recursive clauses, and both pictures are justified by the order-type lemmas rather than left as pictures.

Facts & Assumptions

Given: The ordinals with the operations of Ordinal addition $α + β$ and Ordinal multiplication $α \cdot β$ , and $ω$ the least limit ordinal ( $ω$ is the least limit ordinal, The natural numbers $N$ (von Neumann)).

[L1]

$α \cdot 0 = 0$ , $α \cdot δ^{+} = α \cdot δ + α$ , and $α \cdot λ = ⋃ {α \cdot ξ : ξ \in λ}$ for limit $λ$ (Ordinal multiplication $α \cdot β$ ); $α + 0 = α$ (Ordinal addition $α + β$ ).

[L2]

From Monotonicity of ordinal $+$ and $\cdot$ : strictly increasing and continuous in the right argument, weakly increasing in the left, with left cancellation, and the identities $0 + β = β$ and $1 \cdot β = β$ : $1 \cdot μ = μ \cdot 1 = μ$ (claim (a)); $ν < θ$ implies $α + ν < α + θ$ (claim (b)); $μ \leq ν$ implies $μ γ \leq ν γ$ (claim (e)).

[L3]

For $m, n \in ω$ the ordinal $m \cdot n$ lies in $ω$ (claim (a) of On $ω$ the ordinal $+$ and $\cdot$ are the Peano operations: $ω$ is closed under ordinal $+$ , $\cdot$ and exponentiation, and for naturals $m, n$ the ordinal $m + n$ and $m \cdot n$ are the natural-number sum and product).

[L4]

$ω$ is a limit ordinal, so $⋃ ω = ω$ and $0 \in ω$ ( $ω$ is the least limit ordinal, Successor and limit ordinals); every ordinal is transitive, $μ \subseteq ν$ iff $μ \in ν$ or $μ = ν$ , and $μ \notin μ$ (Ordinal (von Neumann), Basic closure properties of ordinals, Trichotomy and well-ordering of the ordinals).

[L5]

$α \cdot β$ is the order type of $α \times β$ under last differences, that is $β$ copies of $α$ ( $α \cdot β$ is the order type of $α \times β$ ordered by last differences, that is $β$ copies of $α$ ); $α + β$ is the order type of a copy of $α$ followed by a copy of $β$ ( $α + β$ is the order type of $α$ followed by $β$ ); order types are unique (Every well-order has a unique order type) and a strictly increasing bijection between total orders is an order isomorphism (Order embedding and order isomorphism).

Verification

technique · direct

1.1

For $n \in ω$ the ordinal $2 \cdot n$ lies in $ω$ by [L3], hence $2 \cdot n \subseteq ω$ by [L4]; and $n = 1 \cdot n \leq 2 \cdot n$ by [L2], since $1 \leq 2$ , hence $n \subseteq 2 \cdot n$ .

L2L3L4

1.2

$ω \cdot 2 = ω \cdot 1^{+} = ω \cdot 1 + ω = ω + ω$ by [L1] and [L2]; and $ω + ω > ω + 0 = ω$ by [L1] and claim (b) of [L2], since $0 \in ω$ .

L1L2L4

2.1

$2 \cdot ω = ⋃ {2 \cdot n : n \in ω}$ by [L1], and this equals $ω$ : it is contained in $ω$ by step 1.1, and it contains $⋃ {n : n \in ω} = ⋃ ω = ω$ by step 1.1 and [L4].

step 1.1L1L4

3.1

The pictures are the order-type lemmas, not extra assumptions: by [L5], $2 \cdot ω$ is the order type of $2 \times ω$ under last differences, which is $ω$ blocks of two, and step 2.1 evaluates that order type as $ω$ ; while $ω \cdot 2$ is the order type of $ω \times 2$ under last differences, which is two blocks of $ω$ , and by [L5] again that is the order type of a copy of $ω$ followed by a copy of $ω$ , namely $ω + ω$ , in agreement with step 1.2.

step 2.1step 1.2L5

4.1

Therefore $2 \cdot ω = ω$ and $ω \cdot 2 = ω + ω \neq ω$ , so the two products differ.

step 2.1step 3.1step 1.2∎

Remarks

Which convention this depends on. Everything above uses the convention fixed in Ordinal multiplication $α \cdot β$ , that the successor clause appends a copy of the left factor on the right. Under the opposite convention the two values are exchanged, and $2 \cdot ω$ would be $ω + ω$ . Both conventions appear in the literature; this library uses the one stated, throughout.

The general statement. That ordinal multiplication is not commutative is FALSE: ordinal multiplication is commutative, whose refutation is the computation of step 2.1 and step 1.2. The related failure of right distributivity, $(1 + 1) \cdot ω \neq 1 \cdot ω + 1 \cdot ω$ , is FALSE: $(β + γ) \cdot α = β \cdot α + γ \cdot α$ for all ordinals and uses the same value $2 \cdot ω = ω$ .

Why the block picture is a proof and not an illustration. $α \cdot β$ is the order type of $α \times β$ ordered by last differences, that is $β$ copies of $α$ says the product is the order type of the block arrangement, so reading a value off the picture is legitimate once the picture is identified with $α \times β$ under last differences. What is not legitimate is reading it off an unlabelled diagram, which is why step 3.1 names the lemma at each use.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

43 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ordinal arithmetic (Wikipedia) (standard reference, not scraped)
T. Jech, Set Theory, 3rd millennium ed., Ch. 2 (Ordinal numbers) (standard reference, not scraped)
R. Moosa, Set Theory course notes (standard reference, not scraped)
Open Logic Project, Open Logic Text (standard reference, not scraped)

2⋅ω=ω2⋅ω=ω while ω⋅2=ω+ωω⋅2=ω+ω, pictured as order types