Definition·D025
Inductive set
A set containing zero and closed under successor.
In words
I is inductive exactly when the empty set belongs to I and for every x, if x belongs to I then the successor of x belongs to I.
Rests onno axioms yet
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A02 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
An inductive set contains
,
,
, and so on: it contains every number reachable from zero by taking successors, and possibly much more. Infinity asserts that at least one inductive set exists; that is the only place the axioms conjure an infinite object. The intersection of all inductive sets is the set of natural numbers, and "inductive" is what gives mathematical induction both its name and its engine.
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