A Miniaturized “Predicativity” – “slow growing” analogues. Stanley S. Wainer 1 (Leeds UK) HAPPY BIRTHDAY GERHARD December 2013. 1 Earlier parts of this work were done jointly with Elliott Spoors and were partially supported by the 2012 Isaac Newton Institute programme: “Syntax and Semantics; the legacy of A. Turing”.
§ 0. Input–Output Theories for Grzegorczyk Hierarchy. ◮ EA ( I ; O ) is a 2-sorted theory with elementary strength. ◮ EA ( I ; O ) ⊂ EA ( I ; O ) + ⊢ E 3 ( x : I ) ↓ . ◮ EA ( I 1 ; O ) + ( I 2 ) + ⊢ E 4 ( x : I 2 ) ↓ . ◮ EA ( I 1 , I 2 , . . . , I k ; O ) + ⊢ E k +2 ( x : I k ) ↓ . ◮ EA ( I 1 , I 2 , . . . I ω ; O ) ∞ ≺ Γ 0 ⊢ E ω ( x : I ω ) ↓ . The Main Principles: (1) Numerical inputs govern induction-length. (2) Values computable from inputs only may be used as input. (3) There may be many “increasingly refined” levels of input.
§ 1. EA ( I ; O ) – Leivant (1995), Ostrin-Wainer (2005), ◮ Quantified numerical “output” variables a , b , c , . . . . ◮ Unquantified “input” variables x , y , z , . . . (constants). ◮ Terms 0 , Succ , + , × , π, π 0 , π 1 , . . . with usual axioms. ◮ “Predicative/bounded/pointwise Induction” up to x : A (0) ∧ ∀ a ( A ( a ) → A ( a + 1)) → ∀ a ≤ xA ( a ). ◮ Define f ( x ) ↓ ≡ ∃ aC f ( x , a ) for some Σ 1 formula C f . ◮ Then EA ( I ; O ) ⊢ f ( x ) ↓ if and only if f is elementary.
EA ( I ; O ) ⊂ EA ( I ; O ) + – Spoors-Wainer (2012) EA ( I ; O ) is not “user-friendly” since composition of functions f : I → O cannot be proved straightforwardly – however Wirz (2005) developed a variety of derived rules showing this. To remedy this, add a Σ 1 -“Reflection Rule” as in Cantini (2002): Σ( � x ) , ∃ aA ( a ,� x ) Σ( � x ) , ∃ yA ( y ,� x ) where the only free parameters are inputs � x . And add I -quantifiers: Γ , A ( x ) Γ , A ( t ( � x )) Γ , ∃ yA ( y ) . Γ , ∀ yA ( y ) Note: the inductions are still restricted to EA ( I ; O ) formulas only. Then if ⊢ f ( x ) ↓ and ⊢ g ( x ) ↓ we can directly prove ∀ yf ( y ) ↓ and (by reflection) ∃ y ( g ( x ) = y ). Therefore EA ( I ; O ) + ⊢ f ( g ( x )) ↓ .
§ 2. EA ( I 1 , I 2 ; O ) = EA ( I 1 ; O ) + ( I 2 ) + . Add to EA ( I 1 ; O ) + a new layer of I 2 –inputs u , v , . . . and a new level of inductions: A (0) ∧ ∀ a ( A ( a ) → A ( a + 1)) → A ( u ) where A is now any EA ( I 1 ; O ) + formula. Then: ◮ EA ( I 1 ; O ) ⊢ 2 x ↓ ◮ EA ( I 1 ; O ) + ⊢ ∀ x ∃ y (2 x = y ) ◮ EA ( I 1 ; O ) + ⊢ ∃ y (2 x a = y ) → ∃ y (2 x a +1 = y ) ◮ EA ( I 1 ; O ) + ( I 2 ) ⊢ ∀ x ∃ y (2 x u = y ) Now add I 2 –quantifier rules and a Σ 1 –reflection rule for I 2 . This allows compositions of the superexponential etc., so EA ( I 1 ; O ) + ( I 2 ) + ⊢ E 4 ( u ) ↓ .
§ 3. Pointwise Transfinite Induction. Usual transfinite induction TI(A, α ) may be written: A (0) ∧ ∀ γ ( A ( γ ) → A ( γ + 1)) ∧ ∀ λ ( ∀ iA ( λ i ) → A ( λ )) → A ( α ) Definition “Weak, pointwise transfinite induction” PTI(A x ,α ): A (0) ∧ ∀ γ ( A ( γ ) → A ( γ + 1)) ∧ ∀ λ ( ∀ i ≤ xA ( λ i ) → A ( λ )) → A ( α ) or A (0) ∧ ∀ γ ( A ( γ ) → A ( γ + 1)) ∧ ∀ λ ( A ( λ x ) → A ( λ )) → A ( α ) . The idea goes back to U. Schmerl (1982). This is enough to define the Slow Growing function G α ( x ) and is equivalent to Bounded Induction “up to” G α ( x ): A (0) ∧ ∀ a ( A ( a ) → A ( a + 1)) → ∀ a ≤ G α ( x ) A ( a ) .
Tree Ordinals α ≺ Γ 0 and their G α ’s. Definition ◮ α ∈ Ω if α = 0 or ∃ β ∈ Ω( α = β + 1) or α : N → Ω. ◮ G : N × Ω → N is given by G n (0) = 0 , G n ( β + 1) = G n ( β ) + 1 . G n ( λ ) = G n ( λ n ) . ◮ ψ : Ω × Ω → Ω is given by ψ α +1 ( β ) = ψ 2 β ψ 0 ( β ) = β +2 β , α ( β ) , ψ λ ( β ) = sup ψ λ n ( β ) . ◮ F 0 ( m ) = m + 2 m , F n +1 ( m ) = F 2 m n ( m ) . Then F ω ( n ) = F n ( n ). Theorem (i) | ψ α ( ω ) | = Veblen φ α (0) for α ≻ 0 . (ii) G n ( ψ α ( β )) = F G n ( α ) ( G n ( β )) . So G n ( ψ α ( α )) = F ω ( G n ( α )) .
PTI ( α ) in EA ( I , .. ; O ) theories. ◮ Recall: For α ≺ ε 0 , G ( α ) ∈ E 3 , but G ( ε 0 ) �∈ E 3 . ◮ Hence: EA ( I ; O ) ⊢ PTI ( α ≺ ε 0 ), but EA ( I ; O ) �⊢ PTI ( ε 0 ). ◮ Thus: � EA ( I ; O ) + � W = ε 0 = φ 1 (0). ◮ Similarly: � EA ( I 1 , I 2 ; O ) + � W = φ 2 (0) etcetera. ◮ And: � EA ( I 1 , . . . , I ω ; O ) ∞ ≺ Γ 0 � W = Γ 0 . ◮ Wainer-Williams (2005): � ID 1 ( I ; O ) � W = φ ε Ω+1 (0) but ID 1 ( I ; O ) ≡ PA . Note: J¨ ager-Probst (2013) and Ranzi-Strahm (2013), SID ν have full (unstratified) numerical induction in the base theory.
§ 4. EA ( I 1 , I 2 , . . . , I ω ; O ) ∞ ≺ Γ 0 . Sequents are: n : I k ; . . . , m : I i ⊢ α Γ with ω ≥ k > . . . > i . Logic Rules are as follows where β ≺ n α ≺ Γ 0 : n : I k ; . . . , m : I i ⊢ β Γ , A ( ℓ ) ( ∃ I i ) n : I k ; . . . , m : I i ⊢ β C ℓ n : I k ; . . . , m : I i ⊢ α Γ , ∃ x ( I i ( x ) ∧ A ( x )) ( ∀ I i ) { n : I k ; . . . , max( m , j ) : I i ⊢ β Γ , A ( j ) } j n : I k ; . . . , m : I i ⊢ α Γ , ∀ x ( I i ( x ) → A ( x )) level( A ) < k and ( ∨ ) , ( ∧ ) and (Cut) as usual, together with Computation Rules: n ; . . . m ′ ⊢ β ( C ) n ; . . . m ⊢ β C m ′ C ℓ ( Ax ) n ; . . . , m ⊢ α C ℓ if ℓ ≤ q ( m ) n ; . . . m ⊢ α C ℓ
Reading Off Bounding Functions. Ordinal assignment is “slow growing”: |{ β : β ≺ n α }| = G n ( α ). Lemma C k then k ≤ q G n (2 α ) ( m ) . If n ; m ⊢ α Theorem (Basic bounding principle) If EA(I;O) + ⊢ f ( x ) ↓ then, by embedding and cut-reduction, there is an α ≺ ε 0 such that for every x := n, n ; − ⊢ α 0 ∃ aC f ( n , a ) . Then ∃ a ≤ kC f ( n , a ) where k = q G n (2 α ) (0) . So f ∈ E 3 . Lemma ( E 4 bounding) Let B 1 ( α, n ) = q G n (2 α ) (0) be the bounding function at level 1 . Then B 2 ( α, n ) = B 1 ( α ) G n (2 α ) ( n ) is the bound at level 2 : n 2 ; n 1 , − ⊢ α C k ⇒ k ≤ B 2 ( α, max( n 2 , n 1 )) . This bound is E 4 -definable. Etcetera.
§ 5. Level ω – Ackermann. Suppressing ordinal bounds, EA ( I 1 , I 2 , . . . , I ω ; O ) + ∞ proves: ∀ x r ∃ y r ( F 2 a r ( x ) = y ) → ∀ x r ∃ y r ( F 2 a +1 ( x ) = y ) r Hence by induction on a , using repeated cuts: k : I r +1 ; ⊢ ∀ x r ∃ y r ( F r ( x ) = y ) → ∀ x r ∃ y r ( F 2 k r ( x ) = y ) By Cut on ∀ x r ∃ y r ( F r ( x ) = y ) using k : I r +1 ⊢ C k : I r , k : I r +1 ⊢ ∃ y r ( F r +1 ( k ) = y ) Then k : I r +1 ⊢ ∃ y r +1 ( F r +1 ( k ) = y ) so ∀ x r +1 ∃ y r +1 ( F r +1 ( x ) = y ). r : I ω ⊢ ∀ x r ∃ y r ( F r ( x ) = y ). So r : I ω ⊢ ∃ y ω ( F ω ( r ) = y ). But note: r : I ω ⊢ C r : I r . Therefore
§ 6. “Predicativity” in EA ( I 1 , . . . , I ω ; O ) ∞ ≺ Γ 0 . ◮ Sequents are: n : I k ; .. m : I i ⊢ α Γ where ordinal bounds α ≺ Γ 0 are autonomously generated according to the rule: PTI ( β ) ⇒ PTI ( ψ β ( β )). ◮ This holds because if G n ( β ) is computable in the system, so is G n ( ψ β ( β )) = F G n ( β ) ( G n ( β )) = F ω ( G n ( β )) . Only finite iterations of F ω are possible, so can’t reach Γ 0 . ◮ Collapsing Principle: n : I ω ; m : I i ⊢ α m : I i ; ⊢ a C k ⇒ C k where a = G n ( ψ α ( α )) = F ω ( G n ( α ). Computational bounds are finite F -terms, elementary in F ω . ◮ ≺ Γ 0 ⊢ E ω ( x : I ω ) ↓ EA ( I 1 , . . . I ω ; O ) ∞ � EA ( I 1 , . . . I ω ; O ) ∞ ≺ Γ 0 � W = Γ 0 .
References 1) A. Cantini : “Polytime, combinatory logic and positive safe induction”. Archive for Math. Logic Vol. 41 (2002) 169-189. 2) G. J¨ ager & D. Probst : “A proof theoretic analysis of theories for stratified inductive definitions”, new, to appear. 3) D. Leivant : “Intrinsic theories and computational complexity”. In D. Leivant (Ed) LCC’94, LNCS Vol. 960 (1995) 177-194. 4) G. Ostrin & S. Wainer : “Elementary arithmetic”. Annals of Pure and Applied Logic Vol. 133 (2005) 275-292. 5) F. Ranzi & T. Strahm : “A note on the theory SID <ω of stratified induction”, new, to appear. 6) H. Schwichtenberg & S. Wainer : “Proofs and Computations”. ASL Perspectives in Logic, CUP (2012) 465 + xiii. 7) E. Spoors & S. Wainer : “A hierarchy of ramified theories below PRA”. To appear in a volume in honour of Helmut Schwichtenberg, Ontos Math. Logic (2012).
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