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.
Primitive recursive functions
Definition
A total function is primitive recursive when it belongs to the smallest collection of total arithmetic functions that:
- contains every initial arithmetic function from The zero, successor, and projection functions on the natural numbers, and
- is closed under the two schemes from Composition and primitive recursion as closure schemes on arithmetic functions.
Equivalently, a primitive recursive function is one obtainable from the zero, successor, and projection functions by finitely many applications of composition and primitive recursion.
Remarks
-
Primitive recursive functions are always total on .
-
The definition is extensional: different constructions may yield the same primitive recursive function.
Depends on
Used by
- Unbounded minimization and partial recursive functions Definition
- Basic arithmetic operations and bounded quantification are primitive recursive Proposition
- The Ackermann function is total computable but not primitive recursive Proposition
- The sequence-length and coordinate-reading functions are primitive recursive 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
- Jeremy Avigad and Richard Zach, Recursive Functions (standard reference, not scraped)
- Klaus Sutner, Primitive Recursion (standard reference, not scraped)