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Unformatted text preview: 1 FROMAL STATEMENTS OF GODEL'S SECOND
INCOMPLETENESS THEOREM
by
Harvey M. Friedman
January 14, 2007
Abstract. Informal statements of Gödel's Second
Incompleteness Theorem, referred to here as Informal Second
Incompleteness, are simple and dramatic. However, current
versions of Formal Second Incompleteness are complicated
and awkward. We present new versions of Formal Second
Incompleteness that are simple, and informally imply
Informal Second Incompleteness. These results rest on the
isolation of simple formal properties shared by consistency
statements. Here we do not address any issues concerning
proofs of Second Incompleteness.
1. SECOND INCOMPLETENESS FOR PA.
We start with the most commonly quoted form of Gödel's
Second Incompleteness Theorem  for the system PA = Peano
Arithmetic.
PA can be formulated in a number of languages. Of these,
L(prim) is the most suitable for supporting formalizations
of the consistency of Peano Arithmetic.
We write L(prim) for the language based on 0,S and all
primitive recursive function symbols. We let PA(prim) be
the formulation of Peano Arithmetic for the language
L(prim). I.e., the nonlogical axioms of PA(prim) consist of
the
axioms for successor, primitive recursive defining
equations, and the induction scheme applied to all formulas
in L(prim).
INFORMAL SECOND INCOMPLETENESS (PA(prim)). Let A be a
sentence in L(prim) that adequately formalizes the
consistency of PA(prim), in the informal sense. Then
PA(prim) does not prove A.
We have discovered the following result. We let PRA be the
important subsystem of PA(prim), based on the same language
L(prim), where we require that the induction scheme be
applied only to quantifier free formulas of L(prim). 2
FORMAL SECOND INCOMPLETENESS (PA(prim)). Let A be a
sentence in L(prim) such that every equation in L(prim)
that is provable in PA(prim), is also provable in PRA + A.
Then PA(prim) does not prove A.
NOTE: We are referring to equations which may contain
variables (open equations).
We can think of the condition in the above Formal Second
Incompleteness (PA(prim)) as a formal adequacy condition on
a formalization in L(prim) of the consistency of PA(prim).
The following Thesis provides the needed connection between
Formal and Informal Second Incompleteness (PA(prim)).
INFORMAL THESIS (PA(prim)). Let A be a sentence in L(prim)
that adequately formalizes the consistency of PA(prim), in
the informal sense. Then every equation in L(prim) that is
provable in PR(prim), is also provable in PRA + A.
Informal Proof: Let Con(PA(prim)) be an adequate
formalization in L(prim) of the consistency of PA(prim), in
the informal sense. Assume
1) s(x1,...,xk) = t(x1,...,xk) is provable in PA(prim)
where s(x1,...,xk),t(x1,...,xk) are terms in L(prim) whose
variables are among x1,...,xk, k ≥ 0.
We now argue in PRA + Con(PA(prim)). Let x1,...,xk in N, and
assume not s(x1,...,xk) = t(x1,...,xk). Let xi^ be the
numeral for xi; i.e., S...S0, where there are xi S's. Then,
according to PRA,
2) PRA proves not s(x1^,...,xk^) = t(x1^,...,xk^).
In particular,
3) PA(prim) proves not s(x1^,...,xk^) = t(x1^,...,xk^).
From 1) and 3), we see that PA(prim) is inconsistent. This
contradicts Con(PA(prim)). QED
From the Informal Thesis (PA(prim)), we immediately see
that Formal Second Incompleteness (PR(prim)) implies
Informal Second Incompleteness (PR(prim)). 3
Previous attempts to give a Formal Second Incompleteness
(PR(prim)) through a formal adequacy condition were
comparatively unsatisfactory. There were two existing
approaches.
One is through the Hilbert Bernays derivability conditions
and its variants.
These do not provide a direct condition on a formalization
in L(prim) of the consistency of PA(prim). Instead, a proof
predicate is introduced, as well as some auxiliary
syntactic notions, and conditions are placed on all of
these notions, simultaneously. A fully rigorous development
is rather subtle
and involved, and uses the construction of auxiliary
sentences.
Another is the approach of S. Feferman, which places
conditions on the formalization of all the relevant
syntactic notions that lead up to the formalization in
L(prim) of the consistency of PA(prim).
These are more straightforward than the Hilbert Bernays
conditions, but significantly more complicated than the
Hilbert Bernays conditions. They are significantly less
complicated and ad hoc than any direct formalization in
L(prim) of the consistency of PA(prim).
However, they also remain unacceptably complicated
for use in the statement of this vitally important theorem.
The development presented here does NOT address any issues
concerning the complications and subtleties present in
PROOFS of the Second Incompleteness Theorem. It only
relates to the STATEMENT of Second Incompleteness.
2. MANY SORTED EXTENSIONS OF FULL INDUCTION WITH PRIM.
Let L(many) be many sorted predicate calculus. Here the
sorts, constants, relations, and functions are indexed
using nonnegative integers.
Let L be a fragment of L(many) that contains L(prim). For
definiteness, we require that L(prim) lives in the first
sort. We let the theory PA(L) in L consist of the axioms
for successor, the defining equations for the primitive 4
recursive function symbols of L(prim), and the induction
scheme applied to all formulas of L.
INFORMAL SECOND INCOMPLETENESS (many sorted induction,
prim). Let L be a fragment of L(many) containing L(prim).
Let T be a consistent extension of PA(L) in L. Let A be a
sentence in L that adequately formalizes the consistency of
T, in the informal sense. Then T does not prove A.
FORMAL SECOND INCOMPLETENESS (many sorted induction, prim).
Let L be a fragment of L(many) containing L(prim). Let T be
a consistent extension of PA(L) in L. Let A be a sentence
in L such that every equation in L(prim) that is provable
in T, is also provable in PRA + A. Then T does not prove A.
INFORMAL THESIS (many sorted induction, prim). Let L be a
fragment of L(many) containing L(prim). Let T be a
consistent extension of PA(L) in L. Let A be a sentence in
L that adequately formalizes the consistency of T, in the
informal sense. Then every equation in L(prim) that is
provable in T, is also provable in PRA + A.
We now address the question of just which sentences obey
the conditions in Formal Incompleteness above.
THEOREM 2.1. Let L be a fragment of L(many) containing
L(prim). Let T be an extension of PA(L) in L. Let A be a
sentence in L. The following are equivalent.
i. Every equation in L(prim) that is provable in T, is also
provable in PRA + A.
ii. PRA + A proves the formal consistency of every finite
fragment of T.
Furthermore, if T is recursively axiomatized then the
formal consistency of T, formulated on the basis of an
algorithm for the recursive axiomatization, obeys i,ii. If
T contains all true universally quantified equations in
L(prim), then for all A obeying i,ii, A is refutable in
PRA.
3. GENERAL MANY SORTED THEORIES WITH PRIM.
So far, all of our theorems apply only to contexts in which
the induction scheme for all formulas in the language are
present in the theory. In particular, we have not addressed
finitely axiomatized T. 5
We say that a sentence is in P1(prim) if and only if it is a
universally quantified bounded formula in L(prim).
For n ≥ 0, we write ISn(prim) for the fragment of PA(prim)
where the induction scheme is applied to Sn(prim) formulas
only. I.e., formulas in L(prim) starting with at most n
quantifiers, the first of which is existential, followed by
a quantifier free formula.
The following refutes our previous Formal Second
Incompleteness, if we do not insist on extending induction
with respect to all formulas.
THEOREM 3.1. Let T be a fragment of ISn(prim) containing
PRA. There is a theorem A of T such that every theorem of T
lying in P1(prim) is provable in PRA + A.
The key notion needed for General Second Incompleteness is
the notion of relativization. Let S,T be two theories in
L(many), where every symbol appearing in S also appears in
T (with the same sort information). A relativization of S
in T consists of a formula in L(T) which, provably in T,
defines a nonempty set which contains the constants
appearing in the axioms of S, and which is closed under the
operation symbols appearing in S, and where the result of
restricting the quantifiers present in S to this nonempty
set, is provable in T.
INFORMAL SECOND INCOMPLETENESS (general many sorted, prim).
Let L be a fragment of L(many) containing L(prim). Let T be
a consistent extension of PRA in L. Let A be a sentence in
L that adequately formalizes the consistency of T, in the
informal sense. Then T does not prove A.
FORMAL SECOND INCOMPLETENESS (general many sorted, prim).
Let L be a fragment of L(many) containing L(prim). Let T be
a consistent extension of PRA in L. Let A be a sentence in
L such that every universalized equation in L(prim) with a
relativization in T, is provable in PRA + A. Then
T does not prove A.
INFORMAL THESIS (general many sorted, prim). Let L be a
fragment of L(many) containing L(prim). Let T be a
consistent extension of PRA in L. Let A be a sentence in L
that adequately formalizes the consistency of T, in the
informal sense. Then every universalized equation in
L(prim) with a 6
relativization in T, is provable in PRA + A.
We now address the question of just which sentences obey
the conditions in Formal Incompleteness above.
THEOREM 3.2. Let L be a fragment of L(many) containing
L(prim). Let T be an extension of PRA in L. Let A be a
sentence in L. The following are equivalent.
i. Every universalized equation in L(prim) with a
relativization in T, is provable in PRA + A.
ii. PRA + A proves the formal consistency of every finite
fragment of T.
If T is finitely axiomatized, then condition ii asserts
that PRA + A proves the formal consistency of T.
4. MANY SORTED EXTENSIONS OF FULL INDUCTION WITH EXPARITH.
Let L(arith) be the single sorted language based on
0,S,+,dot. Let L(exparith) be the single sorted language
based on 0,S,+,dot,exp, where exp is the binary exponential
function (where exp(x,0) = 1).
Let IS0(exparith) be the fragment of PA(exparith) where
induction is applied to bounded formulas in L(exparith),
only.
Let IS0(arith) be the fragment of PA(arith) where induction
is applied to bounded formulas in L(arith), only.
INFORMAL SECOND INCOMPLETENESS (many sorted induction,
exparith). Let L be a fragment of L(many) containing
L(exparith). Let T be a consistent extension of PA(L) in L.
Let A be a sentence in L that adequately formalizes the
consistency of T, in the informal sense. Then T does not
prove A.
FORMAL SECOND INCOMPLETENESS (many sorted induction,
exparith). Let L be a fragment of L(many) containing
L(exparith). Let T be a consistent extension of PA(L) in L.
Let A be a sentence in L such that every inequation in
L(exparith) that is provable in T, is also provable in
IS0(exparith) + A. Then T does not prove A.
INFORMAL THESIS (many sorted induction, exparith). Let L be
a fragment of L(many) containing L(exparith). Let T be a
consistent extension of PA(L) in L. Let A be a sentence in
L that adequately formalizes the consistency of T, in the 7
informal sense. Then every inequation in L(prim) that is
provable in T, is also provable in IS0(exparith) + A.
THEOREM 4.1. Let L be a fragment of L(many) containing
L(exparith). Let T be an extension of PA(L) in L. Let A be
a sentence in L. The following are equivalent.
i. Every inequation in L(exparith) that is provable in T,
is also provable in IS0(exparith) + A.
ii. IS0(exparith) + A proves the formal consistency of every
finite fragment of T.
Furthermore, if T is recursively axiomatized then the
formal consistency of T, formulated on the basis of an
algorithm for the recursive axiomatization, obeys i,ii. If
T contains all true universally quantified equations in
L(exparith), then for all such A, IS0(exparith) + A is
inconsistent.
5. GENERAL MANY SORTED THEORIES WITH EXPARITH.
INFORMAL SECOND INCOMPLETENESS (general many sorted,
exparith). Let L be a fragment of L(many) containing
L(exparith). Let T be a consistent extension of
IS0(exparith) in L. Let A be a sentence in L that adequately
formalizes the consistency of T, in the informal sense.
Then T does not prove A.
FORMAL SECOND INCOMPLETENESS (general many sorted,
exparith). Let L be a fragment of L(many) containing
L(exparith). Let T be a consistent extension of
IS0(exparith) in L. Let A be a sentence in L such that every
universalized inequation in L(exparith) with a
relativization in T, is provable in IS0(exparith) + A. Then
T does not prove A.
INFORMAL THESIS (general many sorted, exparith). Let L be a
fragment of L(many) containing L(exparith). Let T be an
extension of IS0(exparith) in L. Let A be a sentence in L
that adequately formalizes the consistency of T, in the
informal sense. Then every universalized inequation in
L(exparith) with a relativization in T, is provable in
IS0(exparith) + A.
THEOREM 5.1. Let L be a fragment of L(many) containing
L(prim). Let T be an extension of IS0(exparith). Let A be as
sentence in L. The following are equivalent.
i. Every inequation in L(exparith) with a relativization in
T, is also provable in IS0(exparith) + A. 8
ii. IS0(exparith) + A proves the formal consistency of every
finite fragment of T.
If T is finitely axiomatized, then condition ii asserts
that
IS0(exparith) + A proves the formal consistency of T.
6. GENERAL MANY SORTED THEORIES WITH ARITH.
Here we need to consider P1(arith) sentences. These are the
result of placing zero or more universal quantifiers in
front of a bounded formula of L(arith).
INFORMAL SECOND INCOMPLETENESS (general many sorted,
arith). Let L be a fragment of L(many) containing L(arith).
Let T be a consistent extension of IS0(arith) in L. Let A be
a sentence in L that adequately formalizes the consistency
of T, in the informal sense. Then T does not prove A.
FORMAL SECOND INCOMPLETENESS (general many sorted, arith).
Let L be a fragment of L(many) containing L(arith). Let T
be a consistent extension of IS0(arith) in L. Let A be a
sentence in L such that every sentence in P1(arith) with a
relativization in T, is provable in IS0(arith) + A. Then T
does not prove A.
INFORMAL THESIS (general many sorted, arith). Let L be a
fragment of L(many) containing L(arith). Let T be an
extension of IS0(arith) in L. Let A be a sentence in L that
adequately formalizes the consistency of T, in the informal
sense. Then every sentence in P1(arith) with a
relativization in T, is provable in IS0(arith) + A. ...
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