S. M. Kim
ω–CONSISTENCY AND LÖB’S THEOREM
Abstract
Despite the provable equivalence between Löb’s Theorem and Gödel’s Second
Incompleteness Theorem, the former and its proof do not explicitly involve any
assumption of consistency, while the latter and its proof do. This paper establishes some logico-mathematical relationship between Löb’s Theorem and ω–
consistency.
For the most part, our notation will be that commonly employed
in mathematical logic. We write F ` S to mean that “S, a sentence,
is a theorem of F , a set of sentences”. ω is {0, 1, 2, . . .}. We say that
a formal system Σ is consistent, con(Σ), if there is no wff S such that
both Σ ` S and Σ `∼ S. A formal system Σ is called ω–consistent,
ω–con(Σ), if for every wff S, whenever Σ ` S(n) for every n ∈ ω, then
Σ 6`∼ ∀xS(x). The notion of ω–consistency, which Gödel introduced, allows us to replace semantical arguments by ones that only involve the notion
of provability, which has constructive content. Recall that ω–consistency
implies consistency, but not conversely, as it can be shown as follows: Assume ω–con(Σ) and consider any predicate F(x) such that Σ ` F(x), e.g.,
F(x) ≡ (x = x → x = x). Then Σ ` (n = n → n = n, ∀n ∈ ω). From
ω–con(Σ), we have Σ 6`∼ ∀x(x = x → x = x). Thus con(Σ). To prove that
the converse does not hold, assume Σ to be a consistent extension of Peano
Arithmetic (PA). Then con(Σ ∪ {∼ G}) and ∼ ω–con(Σ ∪ {∼ G}), where
G is the Gödel sentence for Σ.
Remark 1. A version of Σ ∪ {∼ G} was first discussed by Gödel in [1].
He did not prove that it is incomplete.
A proof of the incompleteness of PA can be based on the assumption
that any sentence provable in PA is true. Gödel’s original proof of the First
Incompleteness Theorem (G1) involved the metamathematically stronger
assumption of ω–consistency. Rosser’s incompleteness proof [5], based on
158
the much weaker metamathematical assumption of consistency, allows us
to generate infinitely many distinct formally undecidable sentences in the
consistent formal system in question. As the further advantage of Rosser’s
method over Gödel’s, the former shows incompleteness where the latter
does not.
The undecidability of consistency does not follow from the assumption
of consistency, as we cannot prove that the unprovability of inconsistency
follows from consistency. [Proof: If (provability of inconsistency → inconsistency) is a theorem, then by Löb’s Theorem (LT), inconsistency would
be also a theorem, a contradiction. Hence, (consistency → unprovability of inconsistency) is no theorem.] However, Rosser’s method can prove
the existence of sentences whose undecidability (and hence incompleteness)
follows from consistency.
In spite of the provable equivalence between LT and Gödel’s Second
Incompleteness Theorem (G2) (the proofs due to Kripke, Smoryński and
Kreisel/Lévy [4], [6], [7], [3]), LT or its proof do not explicitly involve any
assumption of consistency, while G2 and its proof do. As the proof of G2 is
largely a formalization of the proof of G1, G2 might be said to have provided
for G1 a cross check on proposed consistency proofs, despite Gödel’s remark
on the irrelevance of G2 to any sensible consistency problem. (As G. Kreisel
in [2] stated: if con(F) is in doubt, why should it be proved in F and not
in an incomparable system?)
Now, it is natural to raise the following questions: Does there exist
any logico-mathematical relationship between LT and consistency (or ω–
consistency)? If so, what is the relationship? In order to characterize our
problem more specifically, we start with:
Theorem 2. Let Σ be any formal system adequate for recursive number
theory. Then ∀m ∈ ω∀n ∈ ω (Fn: a sentence inconsistent with Σ → Σ `
F m ≡ F n).
Proof. Since F m and F n are each inconsistent with Σ, Σ `∼ F m and
Σ `∼ F n. Hence Σ `∼ F m ≡∼ F n, and Σ ` F m ≡ F n.
The problem of the present research can now be defined as the following: What can happen to Theorem 2 if we replace the condition of
159
inconsistency with the weaker condition of ω–inconsistency? The present
paper will be devoted to answering this question.
Definition 3.
(1)
d e
is the Gödel numbering that carries formulas to formal numer-
als.
(2) Following Gödel’s procedure, a Σ1 formula P r(x)(≡ (∃y)yBx)
can be constructed and be called a standard provability predicate for Σ,
any formal system adequate for recursive number theory, iff it satisfies the
following conditions: (for all sentences A and B of the language of Σ)
(i) Σ ` A → Σ ` P r(d Ae )
(ii) Σ ` P r(d A → B e ) → (P r(d Ae ) → P r(d B e ))
(iii) Σ ` P r(d Ae ) → P r(d P r(d Ae )e )
(3) C0 ≡ (0 = f 0) denoting logical falsehood, where f is successor
function.
e
∀n ∈ ω, Cn+1 = P r(d Cn )
e
= P r(d . . . P r(d C0 ) . . .e )
(n + 1 nested P r(d e )s)
e
(4) Con ≡∼ P r(d C0 ) =∼ C1
(5) Σ(0) = PA,
∀n ∈ ω, Σ(n + 1) = Σ(n) ∪ {Σ(n) 6` C0 }
(e.g., Σ(1) = PA ∪{Con})
Fact 4. ∀n ∈ ω − {0}, ∼ Cn ≡ ∧{(Ck+1 → Ck )|k ∈ ω, 0 ≤ k < n}
≡ (Σ(n − 1) 6` C0 ).
Proof. First equivalence: ∧{(Ck+1 → Ck )|k ∈ ω, 0 ≤ k < n}
= (Cn → Cn−1 ) ∧ (Cn−1 → Cn−2 ) ∧ . . . ∧ (C2 → C1 ) ∧ (C1 → C0 )
= (Cn → C0 )
≡∼ Cn , ∀n ∈ ω − {0}.
Second equivalence: ∼ Cn is true (proof by induction on n) and false
sentence C0 is no theorem of Σ(n − 1), ∀n ∈ ω − {0}.
Theorem 5. ∀n ∈ ω, ∼ ω–con(Σ ∪ {Cn }).
160
Proof. It suffices to prove that ω–con(Σ) → Σ 6` Cn , ∀n ∈ ω.
Case k = 0:
Trivially, Σ 6` C0
Case k = n:
Assume Σ 6` Cn . Then Σ `∼ iB“Cn ”, ∀i ∈ ω
Case k = n + 1:
If Σ ` Cn+1 ≡ P r(“Cn ”) ≡ (∃y)yB“Cn ”, then ∼ ω–con(Σ).
Theorem 6. ω–con(Σ) → Σ 6` (Cn+1 → Cn ), ∀n ∈ ω.
Proof.
(1) Σ ` (Cn+1 → Cn ), ∀n ∈ ω
(2) Σ ` Cn , ∀n ∈ ω
(3) ∼ ω–con(Σ)
; Assumption
; LT
; Theorem 5
Theorem 7. Σ ` (Cn → Cn+1 ), ∀n ∈ ω.
Proof. Case n = 0:
Trivially, Σ `∼ C0
Case n ∈ ω − {0}:
Deducible from the third condition of Definition 3 (2).
Thus we have established as an answer to the previously stated question:
Corollary 8. LT → ∃{Cn |n ∈ ω}∀n(n ∈ ω → (∼ ω–con(Σ∪{Cn })∧Σ 6`
(Cn ≡ Cn+1 ))).
Proof. Definition 3(3), Theorem 5, Theorem 6, and Theorem 7.
Acknowledgement. The author is pleased to acknowledge a grant from
the Bloch Foundation, he also would like to thank the logicians at the
University of Bonn UC Berkeley as well as Otto Wilhelms Carls University
for discussions which brought inspiration to write the paper.
References
[1] K. Gödel, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatsh. Math. Phys., 38 (1931),
pp. 173–198, transl. in Collected works, vol. I, Oxford University Press,
(1986), pp. 145–195.
161
[2] G. Kreisel, About Logic and Logicians, Mathematics. A palimpsest
of essays by Georg Kreisel, selected and arranged by Piergiorgio Odifreddi,
vol. II, manuscript (1993).
[3] G. Kreisel and A. Lévy, Reflection principles and their use for establishing the complexity of axiomatic systems, in: Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 14, (1968), pp.
97–142.
[4] S. Kripke, A talk at UCLA Set Theory Meeting (1967).
[5] J. B. Rosser, Extension of some theorems of Gödel and Church,
J. of Symbolic Logic, vol. 1 (1936), pp. 87–91.
[6] C. Smoryński, The incompleteness theorems, in: Handbook of
mathematical logic, Amsterdam–New York–Oxford (1977).
[7] C. Smoryński, The development of self-reference. Löb’s theorem,
manuscript (1991).
Department of Mathematics
Yonsei University, Seoul
Korea
e-mail: sangmunk@bubble.yonsei.ac.kr
162