Definition·D074
Finite sequence
A function from some natural number into a set: a list of that set's elements, indexed from 0.
For a set
:
the set of all finite sequences from
. For
, the length of
is
(D012).
In words
A finite sequence from a set is a function whose domain is some natural number and whose values lie in that set: entry 0, entry 1, and so on up to one less than its length.
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
Exists by Power Set and Separation: every
satisfies
for its length
, using
(ω is a transitive set). Write
for the entries of a length-
sequence. The unique sequence of length
, with domain
(D002), is the empty sequence. Concatenation builds longer sequences from shorter ones; this is the basic machinery behind strings, and eventually proofs as lists of formulas.