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A set equipped with a distinguished zero and a unary successor operation.
The second-order Peano axioms are categorical under full second-order semantics. First-order Peano arithmetic is not categorical by the compactness and Löwenheim–Skolem theorems.
wp-peanoopenlogic-set-theoryEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Choose a distinguished element 0 and a unary map .
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-peanoopenlogic-set-theoryEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Require successor injectivity, zero not to be a successor, and the adopted second-order induction axiom.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-peanomilne-algebraic-number-theory