Definition·D094
Sentence
A formula with no free variable: nothing left in it that could vary with the assignment.
For a language
: a formula
is a sentence
In words
A formula has no free variables - exactly when it is a sentence.
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 · A09 (computed from the citation graph, not asserted).
Remarks
By the setup of satisfaction, whether
holds can only ever depend on
for
(D093) - a fact not yet proved on this wiki, but the entire motivation for isolating sentences: for a sentence, this dependence is on the *empty* set of variables, so
does not depend on
at all, and "M models phi" can be asked without reference to any particular assignment. Every closed first-order claim - the sort of statement a mathematician actually asserts, as opposed to an open formula like "x is prime" - is a sentence in this sense.