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Introduction CHAPTER 1 Introduction 1.1. Proof theory Proof theory originated in Hilbert’s programme for the foundations of mathematics. This programme was a reaction to the debate usually referred to as ‘Grundlagenstreit’, in which Brouwer openly doubted certain principles of reasoning. Hilbert on the contrary, tried to show the correctness of parts of mathematics, by means of broadly accepted ni- tary formalistic concepts. His ambitious programme was the beginning of an attempt to reduce mathematics to the art of mechanical manipulation of symbols according to certain rules; to completely formalize mathematics, including the logical steps in mathematical arguments. As an example, instead of considering the construction of the natural num- bers N = {0, 1, 2, 3,...}, he based himself on their axiomatization: the axioms of Peano arithmetic (PA), consisting in the dening properties of addition and multiplication, and the principle of mathematical induction. However, Godel’s rst incompleteness theorem for PA showed that there is a sentence A (in the appropriate formal rst-order language) such that it has no proof (it cannot be deduced from the axioms of PA), but also its negation ¬A has no proof. Yet one of the two holds in the standard model (N;+, ,S,0), so there is a sentence, holding for N, which is nevertheless unprovable. (Below we will see that this sentence must hence be false in another model of PA.) All this said, Hilbert’s programme failed at least to the extent that it had turned out impossible to fully capture the properties of the natural numbers by an axiomatic approach. Godel’s completeness theorem claries the close relation between truth and derivabil- ity. Let us rst illustrate the dichotomy in logic between semantics on the one hand and syntax on the other hand, by the example of group theory (but we can equally well consider PA, or any other theory). A group can be described as a structure (G; ,e,1) consisting of an underlying set G, a binary multiplication on G,aconstante in G and a unary map ()1 on G, by denition satisfying the rst-order sentences of associativity, the unitary law and the inverse law: gt := {∀x,y,z[x (y z)=(x y) z] , ∀x[[x e = x] ∧ [e x = x]] , 1 1 ∀x[[x x = e] ∧ [x x = e]] } Otherwise said: among all structures G =(G; ,e,1) groups are exactly those for which the axioms gt hold: G|=gt.WesayatheoremA (in the rst-order language of group theory) is true (or holds) (w.r.t. group theory) whenever it holds for all groups, in which case we say A is a semantical consequence of gt; on the other hand A is derivable (w.r.t. group theory) if there exists a formal proof from the axioms gt,inwhichcaseA is called a syntactical consequence of gt.Godel’s completeness theorem (for classical predicate 1 2 1. Introduction logic) now states that these notions coincide: (A holds) ∀G [[G|=gt]⇒[G|=A]] ⇐⇒∃ P [P formally proves gt`A](A is derivable) This was a deep result at the time, as the two notions are totally dierent: the rst one is set theoretical, while the latter is rather of a nitary combinatorial nature. In contrast to theories like group theory, in which one tries to capture a large class of non-isomorphic structures, Peano introduced the language of arithmetic with the inten- tion to describe (N;+, ,S,0) up to isomorphism. However, as we saw in the discussion of Godel’s incompleteness theorem, there is a sentence A, holding for N but unprovable from PA. By the completeness theorem, A must be false in some model of PA. We conclude that Peano’s axioms do not characterize a unique structure. There are several formalizations of the notion of a ‘proof’. The three principle types of formalism are Hilbert system, Natural deduction system and Sequent calculus system (see [TS 96]). 1.2. Hilbert systems A Hilbert system is based on axiom schemes and only two rules, viz. modus ponens (→-elimination) and the rule of generalization (∀-introduction): → AAB modus ponens A generalization (under certain conditions) B ∀yA[y/x] A derivation is a nite tree labeled with formulas, such that the immediate successors of anodeC are the premisses of a rule having C as a conclusion. The root of the tree is the conclusion of the whole derivation. The maximal elements (i.e. the leaves of the tree) are required to be axioms. On the one hand there are purely logical axioms which dene the logical connectives, e.g. A→(B→(A∧B)) ; A→(B→A);[A→(B→C)]→[(A→B)→(A→C)] In addition, there will be axioms for the particular theory (for example, the axioms of group theory). The following is an example of a derivation of A → A in a Hilbert system. Axioms are overlined, as they should be considered as conclusions of 0-ary rules. k s A → ((A → A) → A) [A → ((A → A) → A)] → [(A → (A → A)) → (A → A)] k mp A → (A → A) (A → (A → A)) → (A → A) mp A → A 1.3. Natural deduction systems Another formalism is the system of Natural deduction, rst introduced by Gentzen in [Gentzen 35], and later developed by [Prawitz 65]. While the logical connectives in a Hilbert style derivation are characterized by the logical axioms, in a Natural deduction system the deduction rules determine them: in addition to modus ponens and the rule of generalization, there are introduction and elimination rules for each of the logical connectives. Instead of merely deriving a formula A as in the Hilbert system formalism 1.3. Natural deduction systems 3 from logical and additional axioms at the top of our tree, we now deduce A from a set of so-called open assumptions, which we shall make more precise shortly. A deduction in the system of Natural deduction again is a tree of formulas with a unique conclusion C. However, there is an additional ‘linking structure’. Some of the leaves at the top of the tree are ‘linked to’ exactly one rule in the tree. These leaves are called ‘closed’ assumptions; the others are called ‘open’. This is characteristic for Nat- ural deduction: for some of the rules (viz. →-introduction, ∨-elimination, ∃-elimination and reductio ad absurdum), application of them allows certain open assumptions to be- come closed (by xing the ‘link’). Only the remaining open assumptions are the ‘real assumptions’ on the basis of which one is allowed to conclude C. The Hilbert system and the Natural deduction system for a theory are deductively equivalent, in the sense that there is a (Hilbert style) derivation with conclusion C if and only if there is a deduction (in the Natural deduction formalism) with conclusion C and with (open) assumptions belonging to the axioms for the particular theory. As an example, let us consider the ∧-introduction rule of Natural deduction. It states that, given a deduction with conclusion A and one with conclusion B assumptions assumptions , A B we can build a new deduction with conclusion A ∧ B: assumptions assumptions A B ∧I A ∧ B Another rule, the →-introduction rule, states that, given a deduction with conclusion C, we can build a new deduction with conclusion D → C, which allows (but does not oblige) for the open assumptions D being ‘linked’ (and hence becoming closed). With these rules we can form the following deduction, where the linking structure is indicated by the indices 1 and 2. AB A [B]1 [A]2 [B]1 ∧I ∧I ∧I A ∧ B ∧ ∧ A B →I,1 A B →I,1 B → (A ∧ B) B → (A ∧ B) →I,2 A → (B → (A ∧ B)) We thus deduced the logical axiom A → (B → (A ∧ B)) of a Hilbert system from no assumptions. The other way around, this axiom gives rise to the deduction rule ∧I, as the following compound Hilbert style rule illustrates: 9 → → ∧ > A A (B (A B)) = mp ∧ B B → (A ∧ B) I > mp ; A ∧ B 4 1. Introduction 1.4. -Calculus The natural deduction rule ‘reductio ad absurdum’ (RAA) states that whenever ¬A leads to a contradiction, we may conclude A. Logics without this rule are intuitionistic logics (see [TvD 88]). They admit the Brouwer-Heyting-Kolmogorov (BHK) interpreta- tion, explaining by constructive methods what it means to prove a compound statement in terms of what it means to prove its components: a construction D proves A ∧ B if it is apair(D0, D1) consisting of a proof D0 of A and a proof D1 of B; D proves A ∨ B if D is either of the form (0, D0), and D0 proves A, or of the form (1, D1), and D1 proves B; D proves A → B if D is a construction transforming any proof D0 of A into a proof D(D0) of B; ⊥ is a proposition without proof. A formalized version of the BHK-interpretation is given by the so-called -calculus, consisting of ‘terms’ of certain ‘types’ (the latter for the moment being considered as logical formulas). For each type A there is a countably innite supply of variables xA,yA,..., and each of them is by denition a term of that type. Moreover, if t is a term of type B and xA a variable (possibly occurring several → times in B), then xA t is a term of type A B, which closely corresponds to function abstraction, but also to →-introduction; if t is a term of type A → B and s is a term of type A,thents is a term of type B, which closely corresponds to function application, but also to →-elimination.
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