Fast Growing Functions and Arithmetical Independence Results

Transcription

1 Fast Growing Functions and Arithmetical Independence Results Stanley S. Wainer (Leeds UK) Stanford, March 2013

2 1. Intro A Mathematical Incompleteness Are there any genuine mathematical examples of incompleteness? Goodstein Sequences (1944) Kirby and Paris (1982) Take any number a, for example a = 16. Write a in complete base-2, thus a = Subtract 1, so the base-2 representation is a 1 = Increase the base by 1, to produce the next stage a 1 = = 112. Continue subtracting 1 and increasing the base: a,a 1,a 2,a 3,... Example: 16, 112, 1284, 18753, ,... Theorem (Part 1 Goodstein, Part 2 Kirby & Paris) (1) Every Goodstein sequence eventually terminates in 0. (2) But this is not provable in Peano Arithmetic (PA).

3 1.1. The Hardy Functions If, throughout any Cantor Normal Form α ε 0, we replace ω by a large enough n, then we obtain a complete base-n representation. Subtracting 1, and then putting the ω back, one gets a smaller ordinal P n (α). Hence part 1 of the theorem, by well-foundedness. E.G. With α = ω ωω and n = 2 we get a = 2 22 = 16. Then a 1 = and P n (α) = ω ω+1 + ω ω + ω Definition H 0 (n) = n and H α (n) = H Pn(α)(n + 1). Theorem (Cichon (1983)) H α (n) measures the length of any Goodstein sequence on a,n. A proof that all G-sequences terminate says H ε0 is recursive. But H ε0 is not provably recursive in PA. Hence part 2.

4 1.2. The Fast Growing Hierarchy B α := H 2 α H α+β = H α H β. So B 0 = successor and B α+1 = B α B α. Thus, for finite α, B α (n) = n + 2 α. This is the beginning of the (sometimes called Schwichtenberg - Wainer ) Fast Growing Hierarchy: n + 1 if α = 0 B α (n) = B α (B α (n)) if α = α + 1 B αn (n) if α is a limit where {α n } n N is a chosen fundamental sequence to limit α. Theorem {B α } α< T classifies provable recursion in arithmetical theories T, i.e. provides bounds for witnesses of provable Σ 1 formulas.

5 2. Input-Output Arithmetic Definition EA(I;O) has the language of arithmetic, with (quantified, output ) variables a,b,... It has symbols for arbitrary (partial) recursive functions, given by their defining equations. In addition there are numerical constants ( inputs ) x,y,... Basic terms are those built from the constants and variables by successive application of successor and predecessor. Only basic terms are allowed as witnesses in the logical rules for and. The induction axioms are: A(0) a(a(a) A(a + 1)) A(t(x)) where t is a closed, basic term controlling induction-length!

6 2.1. Working in EA(I;O) Definition Write t for a(t = a). Note that if t is not basic one cannot pass directly from t = t to t. But a + 1 is basic, so t t + 1. In general, t A(t) aa(a) and dually for. Example From b + c we then immediately get b + (c + 1). Therefore b + x by Σ 1 -induction up to x. Then b(b + x ). By iterating this x times, we thus obtain b + x 2. Then b(b + x 2 ), b(b + x 3 ). Etc. Exponential requires a Π 2 induction on b(b + 2 c ).

7 Proving b(b + 2 x ) with Π 2 induction - an argument going back to Gentzen Assume b(b + 2 c ). Then, given an arbitrary b, we have: b + 2 c by the assumption, and hence (b + 2 c ) + 2 c by the assumption again. Therefore b(b + 2 c ) b(b + 2 c+1 ), and b(b ) because b + 1 is basic. Therefore IΠ 2 (I;O) b(b + 2 x ). Similarly IΠ 2 (I;O) b(b + 2 p( x) ). Then IΠ 3 (I;O) b(b + 2 2x ) etcetera.

8 2.2. Bounding Σ 1 -Inductions Theorem Witnesses for Σ 1 theorems, proved by Σ 1 -inductions up to x := n, are bounded by B h where h = log n. Proof. Any induction up to x := n can be unravelled, inside EA(I;O), to a binary tree of Cuts (modus ponens) of height log n. If it s a Σ 1 -induction on aa(b,a) and we assume that B h bounds witnesses at height h, then a further Cut: aa(b,a) aa(b,a) aa(b,a) aa(b,a) aa(b,a) aa(b,a) will yield a bound B h+1 = B h B h at height h + 1.

9 2.3. Provable Functions of EA(I;O) Definition A provably recursive function of EA(I;O) is one which is provably defined on inputs, i.e. f (x). Theorem (Ostrin-W. APAL 133, 2005) The provable functions of I Σ 1 (I;O) are those computable in poly-time p(n). The provable functions of I Σ 2 (I;O) are those computable in exp-time 2 p(n). Etcetera, up the Ritchie-Schwichtenberg hierarchy of all elementary functions. (Cf. Leivant s Ramified Inductions (1995) where such characterizations were first obtained. Also Nelson s Predicative Arithmetic (1986).)

10 2.4. Proof Proof. Fix x := n in f (x) say with d nested inductions. Partial cut-elimination yields a free-cut-free proof, so only inductive cuts remain. The height of the proof-tree will be (of the order of) log n d. For IΣ 1 (I;O) the Bounding Theorem applies immediately to give complexity bounds B log n d (0) = log n d = n d. For IΣ 2 one must first reduce all cuts to Σ 1 form by Gentzen cut-reduction, which further increases the height by an exponential, so in that case the complexity bounds will be B 2 log n d(0) = 2 nd.

11 3. Adding an Inductive Definition Definition ID 1 (I;O) is obtained from EA(I;O) by adding, for each positive inductive form F(X,a), a new predicate P, and Closure and Least-Fixed-Point axioms: for each formula A. Example a(f(p,a) P(a)) a(f(a,a) A(a)) a(p(a) A(a)) Associate the predicate N with the inductive form: F(X,a) : a = 0 b(x(b) a = b + 1).

12 3.1. Embedding Peano Arithmetic Theorem If PA A then ID 1 (I;O) A N. Proof. From the axioms one easily deduces: Therefore: A(0) a(a(a) A(a + 1)) a(f(a,a) A(a)) A(0) a(a(a) A(a + 1)) a(n(a) A(a)). Hence Peano Arithmetic is interpreted in ID 1 (I;O) by relativizing quantifiers to N.

13 3.2. Unravelling LFP-Ax by Buchholz Ω-Rule We are still working in the I/O context, so can fix x := n and unravel inductions into iterated Cuts as before. However the resulting ID 1 (I;O)-derivations will be further complicated by the presence of Least-Fixed-Point axioms. These must be unravelled as well, before we can read off bounds. Definition (Buchholz Ω-Rule) λ 0 N(m),Γ 0 h 0 N(m) λ h Γ 1 λ Γ 0,Γ 1 The map h λ h measures the transformation s uniformity. Hence ordinal heights now appear as the infinitary content.

14 3.3. Ω Proves LFP-Ax The gist of it. For the left-hand premise of the Ω-rule choose 0 N(m), N(m). For the right-hand premise, first assume h 0 N(m). Each step of this (direct, cut-free) proof can be mimicked to prove A(m) if we assume that A is progressive. Thus h a(f(a,a) A(a)),A(m). The standard fundamental sequence for ω gives ω h = h. Ω-rule gives ω a(f(a,a) A(a)), N(m),A(m) for every m. Then by ω-rule obtain LFP-Ax with proof-height ω + 3.

15 3.4. Cut Elimination and Collapsing in ID 1 (I;O) As usual, Gentzen-style cut-reduction raises height exponentially. Lemma (Collapsing) If, for fixed input x := n, we have a cut-free derivation α 0 Γ with Γ positive in N, then k 0 Γ where k B α+1(n). Proof. For Ω-rule, assume it holds for the premises α 0 0 N(m),Γ 0 and h 0 N(m),Γ 0 α h 0 Γ. Then for the left premise, h 0 N(m),Γ 0 where h B α0 +1(n). And for the right premise, k 0 Γ where k B α h +1(n). Hence k B αh +1(n) B α (h + 1) B α B α (n) = B α+1 (n).

16 3.5. A New Proof of an Old Theorem Theorem Every Σ 1 theorem of PA has witnesses bounded by B α for some α < ε 0. Proof. Embed as ID 1 (I;O) a(n(a) A(x,a)) with x input. Translate this into ID 1 (I;O) with proof-height ω + k, cut-rank r. Eliminate cuts to obtain proof-height α = 2 r (ω + k) < ε 0. Collapse to obtain h a(n(a) A(x,a)) with h = B α+1 (x). Use original Bounding Theorem to bound witness a below B h (x) B ω B α+1 (x) B α+2 (x).

17 4. Generalizing to ID <ω Williams thesis (Leeds 2004) generalizes the foregoing to theories of finitely iterated inductive definitions ID i (I;O). As the inductive definitions are iterated, the higher levels of Ω-rules needed to unravel them require ordinals in successively higher number-classes Ω 1,Ω 2,...,Ω i. Collapsing (and Bounding) from one level i + 1 down to the one below is then computed in terms of higher-level extensions of the B α hierarchy: ϕ (i) α (β) for α Ω i+1,β Ω i where ϕ(β) starts with β + 1, composes at successors, diagonalizes at big limits, (just like B) and is continuously extended at small limits. The ordinal bound of ID 2 (I;O) is then the Bachmann-Howard: τ 3 = ϕ (1) ϕ (3) ω (ω 2 ) (ω 1 ) (ω).

18 4.1. Bounding Functions for ID <ω and Π 1 1 -CA 0 Define ϕ (k) : Ω k+1 Ω k Ω k by: β + 1 if α = 0 ϕ (k) ϕ (k) γ ϕ (k) γ (β) if α = γ + 1 α (β) = ϕ (k) α β (β) if α = supα ξ (ξ Ω k ) supϕ (k) α ξ (β) if α = supα ξ (ξ Ω <k ) Define τ = supτ i where τ 0 = ω and τ 1 = ϕ (1) ω (ω), τ 2 = ϕ (1) (ω), τ 3 = ϕ (1) (ω),... ω (ω 1 ) ϕ (3) (ω 1 ) ω (ω 2 ) Theorem The proof-theoretic ordinal of ID i is τ i+2. The provably computable functions of Π 1 1 -CA 0 are those computably-bounded by {B α } α τ.

19 5. Links to Independence Results Theorem (Friedman s Miniaturized Kruskal Theorem for Labelled Trees) For each constant c there is a number K(c) so large that in every sequence {T j } j<k(c) of finite trees with labels from a given finite set, and such that T j c 2 j, there are j 1 < j 2 such that T j1 T j2. The embedding must preserve infs, labels, and satisfy a certain gap condition. Lemma The (natural) computation sequence for B τi (n) satisfies the size-bound above, and is a bad sequence, i.e. no embeddings. Corollary For a simple c n we therefore have B τn (n) < K(c n ) for all n. Therefore K is not provable recursive in ID <ω, nor in Π 1 1 -CA 0.

20 5.1. The Computation Sequence for τ n By reducing/rewriting τ n according to the defining equations of the ϕ-functions, we pass through all the ordinals in τ n [n]. Each term is a binary tree with labels n, and each one-step-reduction at most doubles the size of the tree. E.g. with n = 2 the sequence begins: τ 2 ϕ (1) (ω) 2 (ω ϕ(1) 1) 1 ϕ(2) 1 (ω 1) ϕ (1) 0 ϕ(2) 1 (ω 1) (ϕ(1) 0 ϕ(2) 1 (ω 1) ϕ (1) ω 1 ϕ (1) ω 1 ϕ (1) (ω)) ϕ(1) 0 (ω 1) ϕ(1) 1 (ω 1) ϕ(1) 0 ϕ(2) 1 (ω 1) (ω) ϕ(1) 0 ϕ(2) 0 ϕ(2) 1 (ω 1) (ω) 1 (ω 1) (ϕ(1) 1 (ω 1) (ϕ(1) 0 ϕ(2) 1 (ω 1) (ω))) (ω) ϕ(1) ϕ (1) ω 1 ( ) (ϕ(1) ω 1 ( ))... The length of the entire sequence (down to zero) is therefore τ n [n] = G n (τ n ) = B τn 1 (n). Furthermore, the sequence is bad - no term is gap-embeddable in any follower.