False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-29

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FALSE: $β < γ$ implies $β + α < γ + α$

Statement

FALSE. Ordinal addition (Ordinal addition $α + β$ ) is strictly increasing in its left argument:

$β < γ ⟹ β + α < γ + α for all ordinals α, β, γ .$

What is true is the weak inequality $β \leq γ \Rightarrow β + α \leq γ + α$ , which is claim (c) of 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 $1 \cdot β = β$ . The strict version fails already at $β = 0$ , $γ = 1$ , $α = ω$ , so the weak form is best possible. Right cancellation fails with it: $0 + ω = 1 + ω$ with $0 \neq 1$ .

Facts & Assumptions

Given: The ordinals with the operation of Ordinal addition $α + β$ , and $ω$ the least limit ordinal ( $ω$ is the least limit ordinal, Successor and limit ordinals).

[L1]

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

[L2]

$0 + μ = μ$ (claim (a) of 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 $1 \cdot β = β$ ) and $β \leq α + β$ (claim (c) of the same).

[L3]

For $m, n \in ω$ the ordinal $m + n$ 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 $⋃ ω = ω$ ( $ω$ 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); and $0 \in 1$ , so $0 < 1$ .

Refutation

technique · direct

1.1

For every $n \in ω$ the ordinal $1 + n$ lies in $ω$ by [L3], hence $1 + n \subseteq ω$ by [L4]; and $n \leq 1 + n$ by [L2], hence $n \subseteq 1 + n$ .

L2L3L4

1.2

$0 + ω = ω$ by [L2].

L1L2

2.1

$1 + ω = ⋃ {1 + n : n \in ω}$ by [L1], and that union equals $ω$ : it is contained in $ω$ because each $1 + n \subseteq ω$ by step 1.1, and it contains $ω$ because $ω = ⋃ ω = ⋃ {n : n \in ω}$ by [L4] and each $n \subseteq 1 + n$ by step 1.1.

step 1.1L1L4

3.1

So $0 < 1$ while $0 + ω = ω = 1 + ω$ , which refutes the strict inequality and also refutes right cancellation, since $0 \neq 1$ .

step 2.1step 1.2L4∎

Remarks

Why the left argument is the weak side. The recursion of Ordinal addition $α + β$ runs on the right argument, and at a limit it takes a supremum; a finite head placed on the left is swallowed by that supremum. Concretely, prepending finitely many points to a copy of $ω$ gives a copy of $ω$ again. On the right nothing is swallowed, and there the inequality really is strict, which is claim (b) of 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 $1 \cdot β = β$ .

How much can be lost on the left. As much as one likes below the limit: $n + ω = ω$ for every $n \in ω$ , by the same computation as step 2.1 with $1$ replaced by $n$ . So the map $β \mapsto β + ω$ is constant on $ω$ and collapses infinitely many values.

Left cancellation is unaffected. $α + β = α + γ$ still forces $β = γ$ , because addition is strictly increasing in the right argument. The two cancellation laws are not a package, and this item is exactly the difference.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

29 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)

FALSE: β<γβ<γ implies β+α<γ+αβ+α<γ+α