In
classical deductive logic, a consistent
theory is one that does not entail a
contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a
model, i.e., there exists an
interpretation under which all
formulas in the theory are true. This is the sense used in traditional
Aristotelian logic, although in contemporary mathematical logic the term ''
satisfiable'' is used instead. The syntactic definition states a theory
is consistent if there is no
formula such that both
and its negation
are elements of the set of consequences of
. Let
be a set of
closed sentences (informally "axioms") and
the set of closed sentences provable from
under some (specified, possibly implicitly) formal deductive system. The set of axioms
is consistent when
for no formula
.
If there exists a deductive system for which these semantic and syntactic definitions are equivalent for any theory formulated in a particular deductive
logic, the logic is called
complete. The completeness of the
sentential calculus was proved by
Paul Bernays in 1918 and
Emil Post in 1921, while the completeness of
predicate calculus was proved by
Kurt Gödel in 1930, and consistency proofs for arithmetics restricted with respect to the
induction axiom schema were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931). Stronger logics, such as
second-order logic, are not complete.
A consistency proof is a
mathematical proof that a particular theory is consistent. The early development of mathematical
proof theory was driven by the desire to provide finitary consistency proofs for all of mathematics as part of
Hilbert's program. Hilbert's program was strongly impacted by the
incompleteness theorems, which showed that sufficiently strong proof theories cannot prove their own consistency (provided that they are in fact consistent).
Although consistency can be proved by means of model theory, it is often done in a purely syntactical way, without any need to reference some model of the logic. The
cut-elimination (or equivalently the
normalization of the
underlying calculus if there is one) implies the consistency of the calculus: since there is no cut-free proof of falsity, there is no contradiction in general.
Consistency and completeness in arithmetic and set theory
In theories of arithmetic, such as
Peano arithmetic, there is an intricate relationship between the consistency of the theory and its
completeness. A theory is complete if, for every formula φ in its language, at least one of φ or ¬φ is a logical consequence of the theory.
Presburger arithmetic is an axiom system for the natural numbers under addition. It is both consistent and complete.
Gödel's incompleteness theorems show that any sufficiently strong
recursively enumerable theory of arithmetic cannot be both complete and consistent. Gödel's theorem applies to the theories of
Peano arithmetic (PA) and
primitive recursive arithmetic (PRA), but not to
Presburger arithmetic.
Moreover, Gödel's second incompleteness theorem shows that the consistency of sufficiently strong recursively enumerable theories of arithmetic can be tested in a particular way. Such a theory is consistent if and only if it does ''not'' prove a particular sentence, called the Gödel sentence of the theory, which is a formalized statement of the claim that the theory is indeed consistent. Thus the consistency of a sufficiently strong, recursively enumerable, consistent theory of arithmetic can never be proven in that system itself. The same result is true for recursively enumerable theories that can describe a strong enough fragment of arithmetic—including set theories such as
Zermelo–Fraenkel set theory (ZF). These set theories cannot prove their own Gödel sentence—provided that they are consistent, which is generally believed.
Because consistency of ZF is not provable in ZF, the weaker notion is interesting in set theory (and in other sufficiently expressive axiomatic systems). If ''T'' is a
theory and ''A'' is an additional
axiom, ''T'' + ''A'' is said to be consistent relative to ''T'' (or simply that ''A'' is consistent with ''T'') if it can be proved that
if ''T'' is consistent then ''T'' + ''A'' is consistent. If both ''A'' and ¬''A'' are consistent with ''T'', then ''A'' is said to be
independent of ''T''.
First-order logic
Notation
(Turnstile symbol) in the following context of
mathematical logic, means "provable from". That is,
reads: ''b'' is provable from ''a'' (in some specified formal system). See
List of logic symbols. In other cases, the turnstile symbol may mean implies; permits the derivation of. See:
List of mathematical symbols.
Definition
*A set of
formulas in first-order logic is consistent (written
) if there is no formula
such that
and
. Otherwise
is inconsistent (written
).
*
is said to be simply consistent if for no formula
of
, both
and the
negation of
are theorems of
.
*
is said to be absolutely consistent or Post consistent if at least one formula in the language of
is not a theorem of
.
*
is said to be maximally consistent if for every formula
, if
implies
.
*
is said to contain witnesses if for every formula of the form
there exists a
term such that
, where
denotes the
substitution of each
in
by a
; see also
First-order logic.
Basic results
# The following are equivalent:
##
## For all
# Every satisfiable set of formulas is consistent, where a set of formulas
is satisfiable if and only if there exists a model
such that
.
# For all
and
:
## if not
, then
;
## if
and
, then
;
## if
, then
or
.
# Let
be a maximally consistent set of formulas and suppose it contains
witnesses. For all
and
:
## if
, then
,
## either
or
,
##
if and only if
or
,
## if
and
, then
,
##
if and only if there is a term
such that
.
Henkin's theorem
Let
be a
set of symbols. Let
be a maximally consistent set of
-formulas containing
witnesses.
Define an
equivalence relation on the set of
-terms by
if
, where
denotes
equality. Let
denote the
equivalence class of terms containing
; and let
where
is the set of terms based on the set of symbols
.
Define the
-
structure over
, also called the term-structure corresponding to
, by:
# for each
-ary relation symbol
, define
if
# for each
-ary function symbol
, define
# for each constant symbol
, define
Define a variable assignment
by
for each variable
. Let
be the term
interpretation associated with
.
Then for each
-formula
:
if and only if
Sketch of proof
There are several things to verify. First, that
is in fact an equivalence relation. Then, it needs to be verified that (1), (2), and (3) are well defined. This falls out of the fact that
is an equivalence relation and also requires a proof that (1) and (2) are independent of the choice of
class representatives. Finally,
can be verified by induction on formulas.
Model theory
In
ZFC set theory with classical
first-order logic, an inconsistent theory
is one such that there exists a closed sentence
such that
contains both
and its negation
. A consistent theory is one such that the following
logically equivalent conditions hold
#
[according to De Morgan's laws]
#
See also
*
Equiconsistency
*
Hilbert's problems
*
Hilbert's second problem
*
Jan Łukasiewicz
*
Paraconsistent logic
*
ω-consistency
*
Gentzen's consistency proof
*
Proof by contradiction
Footnotes
References
* 10th impression 1991.
*
*
* (pbk.)
*
*
*
External links
*
{{Authority control
Category:Proof theory
Category:Hilbert's problems
Category:Metalogic