G\"odel's second incompleteness theorem is proved for Herbrand consistency of some arithmetical theories with bounded induction, by using a technique of logarithmic shrinking the witnesses of bounded formulas, due to Z. Adamowicz [Herbrand consistency and bounded arithmetic, \textit{Fundamenta Mathematicae} 171 (2002) 279--292]. In that paper, it was shown that one cannot always shrink the witness of a bounded formula logarithmically, but in the presence of Herbrand consistency, for theories ${\rm I\Delta_0+\Omega_m}$ with $m\geqslant 2$, any witness for any bounded formula can be shortened logarithmically. This immediately implies the unprovability of Herbrand consistency of a theory $T\supseteq {\rm I\Delta_0+\Omega_2}$ in $T$ itself. In this paper, the above results are generalized for ${\rm I\Delta_0+\Omega_1}$. Also after tailoring the definition of Herbrand consistency for ${\rm I\Delta_0}$ we prove the corresponding theorems for ${\rm I\Delta_0}$. Thus the Herbrand version of G\"odel's second incompleteness theorem follows for the theories ${\rm I\Delta_0+\Omega_1}$ and ${\rm I\Delta_0}$.
Deep Dive into Herbrand Consistency of Some Arithmetical Theories.
G"odel’s second incompleteness theorem is proved for Herbrand consistency of some arithmetical theories with bounded induction, by using a technique of logarithmic shrinking the witnesses of bounded formulas, due to Z. Adamowicz [Herbrand consistency and bounded arithmetic, \textit{Fundamenta Mathematicae} 171 (2002) 279–292]. In that paper, it was shown that one cannot always shrink the witness of a bounded formula logarithmically, but in the presence of Herbrand consistency, for theories ${\rm I\Delta_0+\Omega_m}$ with $m\geqslant 2$, any witness for any bounded formula can be shortened logarithmically. This immediately implies the unprovability of Herbrand consistency of a theory $T\supseteq {\rm I\Delta_0+\Omega_2}$ in $T$ itself. In this paper, the above results are generalized for ${\rm I\Delta_0+\Omega_1}$. Also after tailoring the definition of Herbrand consistency for ${\rm I\Delta_0}$ we prove the corresponding theorems for ${\rm I\Delta_0}$. Thus the Herbrand version of
By Gödel's first incompleteness theorem Truth is not the same as Provability in sufficiently strong theories. In other words, Provable is a proper subset of True, and thus True is not conservative over Provable. It is not even Π 1 -conservative; i.e., there exists a Π 1 -formula, in theories which can interpret enough arithmetic, which is true but unprovable in those theories. Thus one way of comparing the strength of a theory T over one of its sub-theories S is considering the Π 1 -conservativeness of T over S. And Gödel's second incompleteness theorem provides such a Π 1 -candidate: Con(S), the statement of the consistency of S. By that theorem S Con(S), but if T Con(S) then T is not Π 1 -conservative over S.
Examples abound in mathematics and logic: Zermelo-Frankel Set Theory ZFC is not Π 1 -conservative over Peano’s Arithmetic PA, because ZFC Con(PA) but PA Con(PA). Inside PA the Σ n -hierarchy is not a Π 1 -conservative hierarchy, since IΣ n+1 Con(IΣ n ) though IΣ n Con(IΣ n ); see e.g. [7]. Then below the theory IΣ 1 things get more complicated: for Π 1 -separating I∆ 0 + Exp over I∆ 0 the candidate Con(I∆ 0 ) does not work, because I∆ 0 + Exp Con(I∆ 0 ). For this Π 1 -separation, Paris and Wilkie [10] suggested the notion of cut-free consistency instead of usual -Hilbert style -consistency predicate. Here one can show that I∆ 0 + Exp CFCon(I∆ 0 ), and then it was presumed that I∆ 0 CFCon(I∆ 0 ), where CFCon stands for cut-free consistency. But this presumption took a rather long time to be established. Meanwhile, Pudlák in [11] established the Π 1 -separation of I∆ 0 + Exp over I∆ 0 by other methods, and mentioned the unprovability of CFCon(I∆ 0 ) in I∆ 0 as an open problem. This problem is interesting in its own right. Indeed Gödel’s second incompleteness theorem has been generalized to all consistent theories containing Robinson’s Arithmetic Q, in the case of Hilbert consistency; see [7]. But for cut-free consistency it is still an open problem whether the theorem holds for Q, and its not too strong extensions. This is a double strengthening of Gödel’s second incompleteness theorem: weakening the theory and weakening the consistency predicate. Let us note that since cut-free provability is stronger than usual Hilbert provability (with a super-exponential cost), then cut free consistency is a weaker notion of consistency. Indeed, proving Gödel’s second incompleteness theorem for weak notions of consistencies in weak arithmetics turns out to be a difficult problem. We do not intend here to give a thorough history of this ongoing research area, let us just mention a few results:
• Z. Adamowicz was the first one to demonstrate the unprovability of cut free consistency in bounded arithmetics, by proving in an unpublished manuscript in 1999 (later appeared as a technical report [1]) that the tableau consistency of I∆ 0 + Ω 1 is not provable in itself. Later with P. Zbierski (2001) she proved Gödel’s second incompleteness theorem for Herbrand consistency of I∆ 0 + Ω 2 (see [2]), and a bit later she gave a model theoretic proof of it in 2002; see [3].
• D. E. Willard introduced an I∆ 0 -provable Π 1 -formula V and showed that any theory whose axioms contains Q + V cannot prove its own tableaux consistency. He also showed that tableaux consistency of I∆ 0 is not provable in itself, see [14,15]; this proved the conjecture of Paris and Wilkie mentioned above.
• S. Salehi (see [13] Chapter 3 and also [12]) showed the unprovability of Herbrand consistency of a re-axiomatization of I∆ 0 in itself, the proof of which was heavily based on [2]. The re-axiomatization used PA -, the theory of the positive fragment of a discretely ordered ring, as the base theory, instead of Q, and assumed two I∆ 0 -derivable sentences as axioms. Also the model-theoretic proof of Z. Adamowicz in [3] was generalized to the I∆ 0 + Ω 1 case in Chapter 5 of [13]. A polished and updated proof of it appears in the present paper.
• L. A. Ko lodziejczyk showed in [8] that the notion of Herbrand consistency cannot Π 1 -separate the hierarchy of bounded arithmetics (this Π 1 -separation is still an open problem). Main results are the existence of an n for any given m 3 such that S m HCon(S n m ), and the existence of a natrual n such that m S m HCon(S n
3 ), where HCon stands for Herbrand consistency.
Herbrand Consistency of Some Arithmetical Theories page 3 (of 20)
• Z. Adamowicz and K. Zdanowski have obtained some results on the unprovability of the relativized notion of Herbrand consistency in theories containing I∆ 0 + Ω 1 ; see [4]. Their paper contains some insightful ideas about the notion of Herbrand consistency.
For I∆ 0 + Ω 1 the arguments are rather smoother, in comparison to the case of I∆ 0 . Our proof for the main theorem on I∆ 0 + Ω 1 borrows many ideas from [3], the major difference being the coding techniques and making use of a more liberal definition of Herbrand consistency. The definition of HCon given in [2] and [3
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
