LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)

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.

Well-ordering finite definition codes

Statement

In ZF, fix a well-order $<$ of a set $A$ . Fix a natural-number enumeration of pairs consisting of a membership formula and an allowed finite parameter arity. A valid definition code is $(e, a)$ , where e specifies arity n and $a \in A^{n}$ . Order codes first by $e \in ω$ , then lexicographically by their tuples of that fixed arity, using $<$ . This well-orders the valid codes. For nonempty A every member of Def(A) has a unique least defining code. For empty A use instead one designated code decoding to empty.

Facts & Assumptions

Given: ZF; a supplied well-order of A and fixed coded formula/arity enumeration. Finite-coordinate minimization establishes the well-order; formula-first ordering avoids the variable-length lexicographic defect.

[F1]

Definable subsets of a membership structure: Def subsets are decoded from formulas and finite parameter tuples; Def(empty) is treated separately.

[F2]

Transfinite induction: Induction on well-orders is available without Choice, in particular on the finite arities.

Proof

1.1

For arity zero the tuple set is the singleton containing the empty tuple. Induct on n: for a nonempty subset of $A^{n + 1}$ , take the least first coordinate occurring in it; its nonempty fibre of n-tuples has a least tuple by induction. Prepending the selected first coordinate gives the lexicographic least member. If A is empty, positive-arity tuple sets are empty and are well-ordered vacuously. Totality and transitivity follow by comparing the first coordinate at which two tuples differ.

F2given

2.1

In any nonempty set of valid codes, first minimize its natural-number e coordinates. The remaining tuples all have the single arity specified by e, and step 1.1 gives a least tuple. This proves the code order is a well-order. For each $b \in Def (A)$ with A nonempty, its decoding fibre is a nonempty set by F1; its least element therefore exists uniquely. For empty A the designated singleton code has the same property. Minimization gives unique representatives, without any appeal to AC.

F1step 1.1∎

Depends on

Used by

HOD as an inner model and comparison with L Theorem
The canonical definable global well-order of L Theorem

Dependency tree · two levels

5 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

Geschke Lemma 5.10 and Exercise 5.11 p17 (standard reference, not scraped)