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.
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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 . Fix a natural-number enumeration of pairs consisting of a membership formula and an allowed finite parameter arity. A valid definition code is , where e specifies arity n and . Order codes first by , 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.
Definable subsets of a membership structure: Def subsets are decoded from formulas and finite parameter tuples; Def(empty) is treated separately.
Transfinite induction: Induction on well-orders is available without Choice, in particular on the finite arities.
Proof
For arity zero the tuple set is the singleton containing the empty tuple. Induct on n: for a nonempty subset of , 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.
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 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.