Herbrand Consistency of I∆0 and I∆0 + Ω1 Saeed Salehi Department of Mathematics University of Tabriz 29 Bahman Boulevard 51666-17766 Tabriz − Iran [email protected] http://saeedsalehi.ir/

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Notation and Basic Definitions

Robinson’s Arithmetic Q is a finitely axiomatized first-order theory that states some very basic facts of arithmetic, like the injectivity of the successor function or the inductive definitions of addition and multiplication. Peano’s arithmetic PA is the first-order theory that extends Q by the induction schema for any arithmetical formula ϕ(x): ϕ(0) & ∀x[ϕ(x) → ϕ(x + 1)] → ∀xϕ(x). Fragments of PA are extensions of Q with the induction schema restricted to a class of formulas. A formula is called bounded if its every quantifier is bounded, i.e., is either of the form ∀x ≤ t(. . .) or ∃ x ≤ t(. . .) where t is a term; they are read as ∀x(x ≤ t → . . .) and ∃x(x ≤ t ∧ . . .) respectively. It is easy to see that bounded formulas are decidable. The theory I∆0 , also called bounded arithmetic, is axiomatized by Q plus the induction schema for bounded formulas. The exponentiation function exp is defined by exp(x) = 2x ; the formula Exp expresses its totality: (∀x∃y[y = exp(x)]). The inverse of exp is log; and the cut log consists of the logarithms of all elements: log = {x | ∃y[exp(x) = y]}. The superscripts above the function symbols indicate the iteration of the functions: exp2 (x) = exp(exp(x)), log2 x = log log x; similarly the cut logn is {x | ∃y[expn (x) = y]}. Let us recall that Exp is not provable in I∆0 ; and sub-theories of I∆0 + Exp are called weak arithmetics. Between I∆0 and I∆0 + Exp a hierarchy of theories is considered in the literature, which has close connections with computational complexity. Let ω0 (x) = x2 and ωn+1 = exp(ωn (log x)) be defined inductively, and let Ωm express the totality of ωm (i.e., Ωm ≡ ∀x∃y[y = ωm (x)]). We have I∆0 + Ωn $ I∆0 + Ωn+1 for every n ≥ 0. Skolemized form of a (preferably prenex normal) formula is obtained by removing the existential quantifiers and replacing their corresponding variables with new (Skolem) function symbols on the universal variables that precedes the quantifier. The resulted universal formula is equi-consistent with the original formula, in the sense that the formula is consistent if and only if the set of instances of its Skolemized form (its Skolem instances) is consistent. This can be generalized to theories (i.e., sets of sentences) as well. Herbrand Consistency of a theory is defined to be the (propositional) consistency of the set of its all Skolem instances. This is a weaker notion than the standard (Hilbert style) consistency, resembling much to cut-free consistency. Thus, for a theory T , Herbrand Consistency, HCon(T ), of T is equivalent to Hilbert Consistency, Con(T ), of T in sufficiently strong theories such as Peano’s Arithmetic PA (or even I∆0 + SuperExp). But in weak arithmetics, like I∆0 or I∆0 + Ω1 these are quite different consistency predicates; which makes it difficult to deal with HCon(−) in them.

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Background History

Two interesting theorems were proved by Z. Adamowicz in [1] about Herbrand Consistency of the theories I∆0 + Ωm for m ≥ 2: Theorem 1. For a bounded formula θ(x) and m ≥ 2, if the theory I∆0 + Ωm + ∃x ∈ logm+1 θ(x) + HConlogm−2 (I∆0 + Ωm ) is consistent, then so is the theory I∆0 + Ωm + ∃x ∈ logm+2 θ(x), where HConlogm−2 (−) is the relativization of HCon(−) to the cut logm−2 .

Theorem 2. For natural m, n ≥ 0 there exists a bounded formula η(x) such that I∆0 + Ωm + ∃x ∈ logn η(x) is consistent, but the theory I∆0 + Ωm + ∃x ∈ logn+1 η(x) is not consistent. These two theorems (by putting n = m + 1 for m ≥ 2) imply together that (∗)

I∆0 + Ωm 6` HConlogm−2 (I∆0 + Ωm ) for m ≥ 2.

This gives a partial answer to the question of holding G¨odel’s Second Incompleteness Theorem for cut-free consistency and weak arithmetics, which was answered in full by D.E. Willard in [4]. The proof of Theorem 1 was adapted to the case of I∆0 + Ω1 in Chapter 5 of [3] by modifying the definition of HCon(−). Thus, one gets another model-theoretic proof of the unprovability of Herbrand Consistency of I∆0 + Ω1 in itself. Note that Theorem 2 holds for I∆0 + Ω1 and I∆0 as well. Later, L.A. Kolodziejczyk extended (∗) above to the following results in [2]: S V Main Theorem (of [2]): There exists n such that the theory m Sm (or, equivalently I∆0 + m Ωm ) does not prove the Herbrand Consistency of S3n (where S3n is defined like S2n but with the language expanded by a function symbol for #3 ). Also the following weaker result is proved in [2]: n ). Theorem 4.1 (of [2]): For every m ≥ 3 there exists an n such that Sm 6` HCon(Sm

These results are taken to show that the Herbrand notion of Consistency cannot serve for Π1 −seperating the theories {I∆0 + Ωk }k or {Sk }k .

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New Results

In the paper version of this extended abstract, we would like to polish the results of [3] (Ch. 5) and make them readable to a wider audience. The newest result is that Theorem 1 can be extended (almost) to I∆0 . Let I be the cut {x | ∃y[y = exp(ω1 2 (x))]} and put T = log I = {x | ∃y[y = exp(x) ∧ y ∈ I ]}. Another way of defining T is {x | ∃y[y = exp2 (x4 )]}. Then we can modify the above theorems of Z. Adamowicz to the following: Theorem 3. For a bounded formula θ(x), if the theory I∆0 + HCon(I40 ) + ∃x ∈ I θ(x) is consistent, then so is the theory I∆0 + ∃x ∈ T θ(x), where I40 is the theory I∆0 augmented with the additional axiom ∀x∃y(y = x · x). Theorem 4. There exists a bounded formula η(x) such that I∆0 + ∃x ∈ I η(x) is consistent, but the theory I∆0 + ∃x ∈ T η(x) is not consistent. 2

These two theorems give rise to a model-theoretic proof of I∆0 6` HCon(I40 ). We note that the proof of Theorem 1 in [1] goes (roughly) as follows: for a model M |= I∆0 + Ωm + ∃x ∈ logm+1 θ(x) + HConlogm−2 (I∆0 + Ωm ) we construct an inner model N |= I∆0 + Ωm (by M |= HConlogm−2 (I∆0 + Ωm )) which is definable in M, and then squeeze a witness for x ∈ logm+1 &θ(x) logarithmically to get a witness for x ∈ logm+2 &θ(x) in N , hence we obtain the desired model N |= I∆0 + Ωm + ∃x ∈ logm+2 θ(x). In the squeezing process, we need to demonstrate a large (non-standard) number with a relatively small code (G¨ odel) number – as small as the logarithm of the number. Thus we added the redundant (and already Q−provable) sentence ∀x∃y(y = x · x) to the set of axioms of I∆0 to get a Skolem function m symbol like f with the interpretation f(x) = x2 . Now we have fm (2) = 22 and we can code this number by O(2m ). This is the reason we could formulate Theorem 3 for I40 instead of I∆0 , and then get its corollary as I∆0 6` HCon(I40 ), though it would have been more desirable to give this kind of proof for the unprovability I∆0 6` HCon(I∆0 ). Also, L.A. Kolodziejczyk notes in [2] that his main theorem could be extended to the case of S2n by either having a function symbol for S ω1 in the language or adding the seemingly irrelevant axiom ∀x∃y(y = x#x). He then mentions that m Sm 6` HCon∗ (S2m + [∀x∃y(y = x#x)]). This goes to say that by having an additional formula in the list of axioms, though it may seem too trivial to be an axiom, one can shorten the cut-free proofs exponentially (cf. Clarification on p. 474 in [4] and Remark on p. 636 in [2]). To see why the proof of Theorem 2 in [1] works for Theorem 4 as well, we note that the proof does not explicitly construct a bounded formula η(x) which satisfies the conditions of the theorem. The proof is by contradiction: assume for all bounded formulas θ(x), the consistency of I∆0 + Ωn + ∃x ∈ logm θ(x) implies the consistency of I∆0 + Ωn + ∃x ∈ logm+1 θ(x), and then we get a contradiction. The essential relevance of the cuts logm and logm+1 in the proof is the equivalence 2x ∈ logm ⇐⇒ x ∈ logm+1 . Indeed, any two cuts I , T which satisfy this equivalence (2x ∈ I ⇐⇒ x ∈ T ) will work in the proof of Theorem 2 in [1]. And our cuts I , T = log I defined above, just have this relation with each other. Finally, we believe that with the modified definition and formalization of Herbrand Consistency in [3], the above mentioned results of [2] can be improved considerably – for the case of S2n and a little beyond. This is yet to be seen.

References 1. Adamowiz, Z.: Herbrand Consistency and Bounded Arithmetic. Fund. Math. 171, 279–292 (2002) http://journals.impan.pl/fm/Inf/171-3-7.html 2. Kolodziejczyk, L.A.: On the Herbrand Notion of Consistency for Finitely Axiomatizable FGragments of Bounded Arithmetic Theories. J. Symb. Log. 71, 624–638 (2006) http://projecteuclid.org/euclid.jsl/1146620163 3. Salehi, S.: Herbrand Consistency in Arithmetics with Bounded Induction. Ph.D. Dissertation in Institute of Mathematics, Polish Academy of Sciences (2002) http://saeedsalehi.ir/pphd.html 4. Willard, D. E.: How to Extend the Semantic Tableaux and Cut-Free Versions of the Second Incompleteness Theorem Almost to Robinson’s Arithmetic Q. J. Symb. Log. 67, 465–496 (2002) http://projecteuclid.org/euclid.jsl/1190150055

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