On fixed points, diagonalization, and self-reference Bernd Buldt - PowerPoint PPT Presentation

Fixed Points Diagonalization Self-Reference On fixed points, diagonalization, and self-reference Bernd Buldt Department of Philosophy Indiana U - Purdue U Fort Wayne (IPFW) Fort Wayne, IN, USA e-mail: buldtb@ipfw.edu CL 16 – Hamburg – September 12, 2016 Fixed Points, Diagonalization, Self-Reference CL 16, Hamburg 2016

Fixed Points Diagonalization Self-Reference 3 Section I: G1 & Fixed Points Fixed Points, Diagonalization, Self-Reference CL 16, Hamburg 2016

Fixed Points Diagonalization Self-Reference 3 G1 Proof, using the G¨ odel fixed point Assumptions (ADQ) ⊢ F ϕ ⇔ ⊢ F Pr F ( � ϕ � ), for all ϕ ∈ L F (FPE) ⊢ F γ ↔ ¬ Pr F ( � γ � ), for at least one γ ∈ L F Proof ADQ FPE con F ⊢ F γ ⇒ ⊢ F ¬ Pr F ( � γ � ) ⇒ ⊢ F ¬ γ ⇒ � ⇒ �⊢ F γ FPE ADQ con F ⊢ F ¬ γ ⇒ ⊢ F ¬ Pr F ( � γ � ) ⇒ ⊢ F γ ⇒ � ⇒ �⊢ F ¬ γ Fixed Points, Diagonalization, Self-Reference CL 16, Hamburg 2016

Fixed Points Diagonalization Self-Reference 3 Fixed point derivation, Step 1: Substitution ◮ Fix a certain individual variable of your choice; say ‘u.’ ◮ Define a function sub that mirrors the substitution of the replacee variable ‘u’ for a replacer term ‘t,’ ϕ [u] t u ≡ ϕ (t), but in the realm of G¨ odel numbers. In short: � gn ( ϕ [u] t u ) if x = gn ( ϕ (u)) and y = gn (t) sub ( x , y ) := otherwise . x ◮ Note that sub ( x , y ) is primitive recursive and therefore represented by an expression ϕ s (x , y) in F . Fixed Points, Diagonalization, Self-Reference CL 16, Hamburg 2016

Fixed Points Diagonalization Self-Reference 3 Fixed point derivation, Step 2: Definitions ◮ Define ϕ (u) : ≡ ∀ x � � ¬ Proof F (x , sub(u , u)) . ◮ Define p := gn ( ϕ (u)). ◮ Substitute p for u in ϕ (u), viz. , γ : ≡ ϕ (p) ≡ ∀ x [ ¬ Proof F ( x , sub (p , p))] . ◮ Calculate � � sub ( p , p ) = gn ( ϕ (u)) , p ; def. p sub ϕ [u] p � � = gn ; def. sub u � � = ϕ (p) ; substitution gn = gn ( γ ) ; def. γ Fixed Points, Diagonalization, Self-Reference CL 16, Hamburg 2016

Fixed Points Diagonalization Self-Reference 3 Fixed point derivation, Step 3: Derivation ◮ Recall Step 2: sub ( p , p ) = gn ( γ ). ◮ Reason inside F . ⊢ F ¬ Pr F (x) ↔ ¬ Pr F (x) ; logic ⊢ F ¬ Pr F (sub(p , p)) ↔ ¬ Pr F ( � γ � ) ; Step 2 � � ⊢ F ∀ x ¬ Proof F ( x , sub (p , p)) ↔ ¬ Pr F ( � γ � ) ; def. Pr F ⊢ F ϕ (p) ↔ ¬ Pr F ( � γ � ) ; def. ϕ (p) ⊢ F γ ↔ ¬ Pr F ( � γ � ) ; def. γ ◮ Warning. We assumed ⊢ F sub(p , p) = � γ � , which requires induction. Fixed Points, Diagonalization, Self-Reference CL 16, Hamburg 2016

Fixed Points Diagonalization Self-Reference 3 Theorem (Fixed Point Theorem, Diagonalization Lemma) Assume F to allow for representation. For each expression ϕ with at least one variable free, there is a ψ such that, ⊢ F ψ ↔ ϕ ψ where ϕ ψ can be either of the four forms: ϕ ( � ψ � ) , ϕ ( � ¬ ψ � ) , ¬ ϕ ( � ψ � ) , ¬ ϕ ( � ¬ ψ � ) , viz., instances of what we call a Henkin, Jeroslov, G¨ odel, or Rogers fixed point resp. Proof. Same as above (with minor modifications). Fixed Points, Diagonalization, Self-Reference CL 16, Hamburg 2016

Fixed Points Diagonalization Self-Reference 3 Black self-referential magic? ◮ Two questions about fixed points such as ⊢ F γ ↔ ¬ Pr F ( � γ � ). 1. How much “black magic” is required for their derivation? . . . will be answered in Section II. 2. How much “self-reference” do they involve? . . . will be answered in Section III. Fixed Points, Diagonalization, Self-Reference CL 16, Hamburg 2016

Fixed Points Diagonalization Self-Reference 4 Section II: Diagonalization Fixed Points, Diagonalization, Self-Reference CL 16, Hamburg 2016

Fixed Points Diagonalization Self-Reference 4 Black magic? 1 st Question How much “black magic” is required for the derivation of fixed points such as ⊢ F γ ↔ ¬ Pr F ( � γ � ) ? Answer None. Fixed Points, Diagonalization, Self-Reference CL 16, Hamburg 2016

Fixed Points Diagonalization Self-Reference 4 Diagonalization ◮ Let A = { a ij } i , j ∈ ω be a (countable) two-dimensional array: R 0 : a 00 a 01 . . . a 0 n . . . R 1 : a 10 a 11 . . . a 1 n . . . . . . ... . . . . . . R n : a n 0 a n 1 . . . a nn . . . . . . ... . . . . . . ◮ Let f be a sequence transforming function, f ( R n ) = { f ( a ni ) } i ∈ ω . ◮ Apply f to the diagonal sequence D : D ′ = f ( D ) := � f ( a 00 ) , f ( a 11 ) , f ( a 22 ) , . . . , f ( a nn ) , . . . � . Fixed Points, Diagonalization, Self-Reference CL 16, Hamburg 2016

Fixed Points Diagonalization Self-Reference 4 Diagonalization: (Non-)Closure ◮ One of two things can happen to the anti-diagonal D ′ = f ( D ): 1. D ′ is identical to one of the rows, viz. , f ( D ) = R i ∈ A , for some i . 2. D ′ is not identical to any of the rows, viz. , f ( D ) � = R i ∈ A , for all i . ◮ If Case 1 applies, we call the set A closed under f , and f will have fixed points. ◮ If Case 2 applies, A is not closed under f , and we have Cantor’s diagonal argument showing that a certain sequence is not in A (to “diagonalize out”). Fixed Points, Diagonalization, Self-Reference CL 16, Hamburg 2016

Fixed Points Diagonalization Self-Reference 4 Diagonalization: Case 1 – Closure ◮ D ′ is identical to one of the rows, viz. , f ( D ) = R i ∈ A , for some i . ◮ The identity D ′ = f ( D ) = R i is element-wise identity: D ′ = � f ( a 00 ) , f ( a 11 ) , . . . , f ( a ii ) , . . . , f ( a nn ) , . . . � � � � � R i = � a i 0 , a i 1 , . . . , a ii , . . . , a in , . . . � ◮ Closure under f (failure to “diagonalize out” ) implies fixed points f ( a ii ) = a ii . Fixed Points, Diagonalization, Self-Reference CL 16, Hamburg 2016

Fixed Points Diagonalization Self-Reference 4 Diagonalization: Case 1 – Closure R 0 : a 00 a 01 a 0 n R 0 : fa 00 a 01 a 0 n . . . . . . . . . . . . R 1 : a 10 a 11 a 1 n R 1 : a 10 fa 11 a 1 n . . . . . . . . . . . . . . . . . . ... ... . . . . . . ⇒ . . . . . . R n : a n 0 a n 1 a nn R n : a n 0 a n 1 fa nn . . . . . . . . . . . . . . . . . . ... ... . . . . . . . . . . . . R 0 : a 00 a 01 a 0 i a 0 n . . . . . . . . . R 1 : a 10 a 11 . . . a 1 i . . . a 1 n . . . . . . . ... . . . . . . . . ⇒ fa 00 fa 11 fa ii fa nn f ( D ) = R i : . . . . . . . . . a i 0 a i 1 a ii a in . . . . ... . . . . . . . . R n : a n 0 a n 1 . . . a ni . . . a nn . . . Fixed Points, Diagonalization, Self-Reference CL 16, Hamburg 2016

Fixed Points Diagonalization Self-Reference 4 Diagonalization: Closure & G¨ odel fixed point ◮ Can we understand γ ↔ ¬ Pr F ( � γ � ) to be an instance of f ( a ii ) = a ii for some f and some array A = { a ij } i , j ∈ ω ? ◮ Yes. Fixed Points, Diagonalization, Self-Reference CL 16, Hamburg 2016

Fixed Points Diagonalization Self-Reference 4 Diagonalization: Closure & G¨ odel fixed points ◮ Step 1: Choose all first-order expressions with the free variable ‘u:’ A = { ϕ 0 (u) , ϕ 1 (u) , ϕ 2 (u) , . . . } . ◮ Step 2: Form the set of all of their G¨ odel numbers: B = { � ϕ 0 (u) � , � ϕ 1 (u) � , � ϕ 2 (u) � , . . . } . ◮ Step 3: Systematically plug all members of B into the free variable slots of all members of A ; call this set C . We write ‘ ϕ ab ’ instead of ‘ ϕ a ( � ϕ b � ).’ Fixed Points, Diagonalization, Self-Reference CL 16, Hamburg 2016

Fixed Points Diagonalization Self-Reference 4 odel fixed points – 1 st diagonalization Diagonalization: G¨ ◮ Lay out the elements of C in such a way that A determines the rows and B the columns which gives us:: � ϕ 0 � � ϕ 1 � � ϕ n � ϕ 0 ϕ 00 ϕ 01 . . . ϕ 0 n . . . ϕ 1 ϕ 10 ϕ 11 . . . ϕ 1 n . . . . . . ... . . . . . . ϕ n ϕ n 0 ϕ n 1 . . . ϕ nn . . . . . . ... . . . . . . ◮ Note that the diagonal sequence { ϕ xx } x ∈ ω corresponds to the substitution function sub ( x , x ) we used above. Fixed Points, Diagonalization, Self-Reference CL 16, Hamburg 2016

Fixed Points Diagonalization Self-Reference 4 odel fixed points – 2 nd diagonalization Diagonalization: G¨ 1. Observe that the provability predicate ¬ Pr F (u) is itself part of the first set we started out with: A = { ϕ 0 , ϕ 1 , ϕ 2 , . . . } ; i. e., ∃ i s. t.: ϕ i ≡ ¬ Pr F ( u ). 2. Apply the transformation f : ϕ ab �→ ¬ Pr F ( ϕ ab ). 3. Because of (1), f maps C onto C , C will be closed under f , and each image ¬ Pr F ( ϕ ab ) must be a ϕ in , for some n . 4. Hence, f ( D ) has a fixed point ϕ ii , which corresponds to the expression γ ≡ ϕ (p) we used above. Fixed Points, Diagonalization, Self-Reference CL 16, Hamburg 2016