Alphabeta MathHow 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.
Transfinite induction
Statement
Let be a well-order (Well-order and well-ordered set) and let satisfy the following: for every , if then (Initial segment of a well-order). Then .
In property form: if a property of elements of satisfies "whenever holds for every , it holds at ", then holds for every .
This is a theorem of ZF. No form of the Axiom of Choice is used. Choice is perfectly available at this point in the library, since Zorn's lemma is proved from it on the previous page; the claim made here is about this proof, which never invokes it.
Facts & Assumptions
Given: A well-order and a subset with the stated closure property. No base case is assumed separately: the hypothesis applied to the least element of , where the initial segment below it is empty, supplies it.
For every : if then .
Every nonempty subset of has a least element, and the order is total (Well-order and well-ordered set).
Proof
Suppose, for contradiction, that .
Since , the set is then nonempty.
Let , the least element of in the order of , which exists because is a nonempty subset of .
Every with satisfies , by minimality of , hence ; that is, .
The hypothesis on therefore applies at and gives .
But gives , which is impossible; hence is empty and .
Remarks
Where the well-ordering is used. Only once, to produce the least element of . Nothing else about enters the argument, so the same proof establishes the statement for an arbitrary well-founded relation once that notion is available.
No separate base case. Applying the hypothesis at , when is nonempty, forces , because the initial segment below is empty and is therefore contained in vacuously. This is why transfinite induction is stated with a single clause where ordinary induction has two, and it is also why the statement holds vacuously for .
Successors and limits. The scheme that looks stronger, with one clause for the least element, one for successors and one for limits, is the specialisation of this theorem to a well-order in which those three kinds of element are distinguished. Nothing is gained by assuming it separately, and the three-clause form is what gets used once ordinals are available.
Least counterexample. The dual reading is often the useful one: if then the hypothesis must fail somewhere, so there is a least with and every in . Steps 2.1 to 5.1 are precisely the construction of that least counterexample.
Depends on
Used by
- The clauses at 0, at a successor and at a limit determine exactly one operation α ↦ ℵ_α, in ZF, and — assuming the Axiom of Choice — exactly one operation α ↦ ℶ_α; each value is an infinite cardinal, each is strictly increasing and continuous at limits, and α ≤ ℵ_α Corollary
- An ordinal α with ℵ_α = α, built as the supremum of the tower ℵ₀, ℵ_ℵ₀, ℵ_ℵ_ℵ₀, …, and its cofinality is ℵ₀ Example
- FALSE: transfinite induction and recursion need the Axiom of Choice False statement
- Borel hierarchy exhaustion and preservation by continuous pullback Lemma
- Countable tree ranks and monotonicity under extension maps Lemma
- Metric Borel hierarchy inclusions and fixed-rank operations Lemma
- Proper iteration master-condition lemma Lemma
- Rigidity of well-orders Lemma
- Subgroups of free abelian groups are free Lemma
- Trichotomy and well-ordering of the ordinals Lemma
- Under Choice, a submodule of an arbitrary-rank free module over a PID is free Lemma
- Well-ordering finite definition codes Lemma
- α · β is the order type of α × β ordered by last differences, that is β copies of α Lemma
- α + β is the order type of α followed by β Lemma
- Transitivity and growth of hierarchy stages Proposition
- A submodule of an arbitrary-rank free module over a PID is free Theorem
- Borel payoffs admit unraveling covers Theorem
- Cantor normal form: every nonzero ordinal is ω^β₀· c₀ + ⋯ + ω^βₖ₋₁· cₖ₋₁ with β₀ > ⋯ > βₖ₋₁ and each cᵢ a nonzero natural number, in exactly one way Theorem
- Every set is countable in the intermediate extension Theorem
- Every successor ordinal is compact in its order topology and every limit ordinal is not; and, assuming countable choice, ω₁ is countably compact and sequentially compact while ω₁ + 1 is compact Theorem
- Every well-order has a unique order type Theorem
- Hessenberg: κ ⊗ κ = κ for every infinite cardinal κ, proved in ZF from the canonical well-order of κ × κ Theorem
- Higher Tor over the integers vanishes Theorem
- Monotonicity of ordinal + and ·: strictly increasing and continuous in the right argument, weakly increasing in the left, with left cancellation, and the identities 0 + β = β and 1 · β = β Theorem
- Ordinal multiplication is associative, and α · (β + γ) = α·β + α·γ Theorem
- Transfinite recursion Theorem
- α^β+γ = α^β·α^γ and (α^β)^γ = α^β·γ; and for α > 1 exponentiation is strictly increasing with β ≤ α^β Theorem
Dependency tree · two levels
6 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
- Transfinite induction (Wikipedia) (standard reference, not scraped)
- Well-order (Wikipedia) (standard reference, not scraped)
- A. Marks, Set Theory (standard reference, not scraped)