Axiomatic WikiWWikiDDefinitionsConsistent theory
Definition·D106
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, writing negation via the usual notation
: 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.