Definition·D106
Consistent theory
A theory is consistent when it never proves both a formula and its negation.
For a language
and
:
is consistent
In words
A theory is consistent exactly when there is no formula it proves along with that formula's negation.
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
The purely syntactic notion of consistency, needing no structure or model - a theory could in principle be inconsistent even if no model has yet been considered. Soundness gives one direction of the expected link to semantics for free: if
has a model
,
is consistent (were
and
both to hold, soundness would give
and
, i.e.
and not
for some assignment
, a contradiction in the ambient logic). The converse - every consistent theory has a model - is the deep content of Gödel's completeness theorem, reached via Lindenbaum's lemma and the Henkin construction.