Fixed Points meet Löb’s Rule Feferman’s G2 plus Examples Uniform Albert Visser Semi-numerability The Henkin Calculus Philosophy, Faculty of Humanities, Utrecht University The µ -Calculus Proof Theory Virtual Seminar, November 18, 2020 1
Overview Feferman’s G2 plus Examples Feferman’s G2 plus Examples Uniform Semi-numerability The Henkin Calculus The µ -Calculus Uniform Semi-numerability The Henkin Calculus The µ -Calculus 2
Overview Feferman’s G2 plus Examples Feferman’s G2 plus Examples Uniform Semi-numerability The Henkin Calculus The µ -Calculus Uniform Semi-numerability The Henkin Calculus The µ -Calculus 2
Overview Feferman’s G2 plus Examples Feferman’s G2 plus Examples Uniform Semi-numerability The Henkin Calculus The µ -Calculus Uniform Semi-numerability The Henkin Calculus The µ -Calculus 2
Overview Feferman’s G2 plus Examples Feferman’s G2 plus Examples Uniform Semi-numerability The Henkin Calculus The µ -Calculus Uniform Semi-numerability The Henkin Calculus The µ -Calculus 2
Overview Feferman’s G2 plus Examples Feferman’s G2 plus Examples Uniform Semi-numerability The Henkin Calculus The µ -Calculus Uniform Semi-numerability The Henkin Calculus The µ -Calculus 3
Feferman’s G2 Feferman’s G2 plus Examples We fix an arithmetisation “proof ( X , x , y ) ” for “ x is a proof of y from Uniform axioms in X ”. Here X is a unary predicate symbol. We write Semi-numerability prov α ( y ) for ∃ x proof ( α, x , y ) . The Henkin Calculus The µ -Calculus Theorem ( ± Feferman 1960) Consider any consistent RE theory T and an interpretation K of EA + B Σ 1 in T. Suppose σ is Σ 1 and σ K semi-numerates the axioms of T in T. Then, we have T � ( con ( σ )) K . One can omit B Σ 1 , but then we need a modification of the proof. 4
Limitations Feferman’s G2 plus Examples ◮ We have G2 for oracle provability, the provability notion Uniform Semi-numerability associated with ω -consistency, cut-free EA-provability over The Henkin EA, etcetera. Calculus ◮ EA + B Σ 1 seems far too strong to be a convincing base The µ -Calculus theory. ◮ The role of the very specific formula-class Σ 1 . We provide a more general Feferman-style result that works for certain predicates of the form prov α that do not satisfy the Löb conditions. 5
The (non-)Role of the Löb Conditions Feferman’s G2 plus Examples Feferman’s proof employs the Löb Conditions for prov K . Uniform Semi-numerability There is a Σ 0 1 -numeration σ of the axioms of EA in EA such that: The Henkin Calculus The µ -Calculus ◮ EA ⊢ ∃ x ∀ y ∈ σ y < x . ◮ EA � σ ⊤ ↔ σ ⊥ . σ ◮ EA ⊢ G ↔ σ ⊥ , for any G with EA ⊢ G ↔ ¬ σ G . σ So the Löb Conditions fail for EA. However, the result , G2 for Σ 1 -semi-numerations, does hold —as follows from result below. 6
Numerability is not Sufficient An Example due to Feferman. Feferman’s G2 plus Examples Uniform Let π be a standard predicate representing the axioms of PA. Let Semi-numerability π x ( y ) : ↔ π ( y ) ∧ y ≤ x . The Henkin Calculus The µ -Calculus We define π ⋆ ( y ) : ↔ π ( y ) ∧ con ( π y ) . Note that π ⋆ is Π 0 1 . π ⋆ numerates the axioms of PA in PA, but we do have PA ⊢ con ( π ⋆ ) . To verify in PA that the first k axioms are indeed axioms we need axioms enumerated after stage k . Thus, the restriction to Σ 1 is needed in Feferman’s Theorem. 7
Overview Feferman’s G2 plus Examples Feferman’s G2 plus Examples Uniform Semi-numerability The Henkin Calculus The µ -Calculus Uniform Semi-numerability The Henkin Calculus The µ -Calculus 8
Uniform Semi-numerability What goes wrong in Feferman’s Example is all that goes wrong. Feferman’s G2 plus Examples Uniform We fix proof ( X , x , y ) . Semi-numerability The Henkin Calculus Consider a consistent theory U . Let X be the set of (Gödel The µ -Calculus numbers of) axioms of U . There are no constraints on the complexity of X . Let U k be axiomatised by X k , the elements of X that are ≤ k . Suppose N : S 1 2 � U . A U -formula ξ uniformely semi-numerates X (w.r.t. N ) iff, for every n , there is an m ≥ n , such that U m proves ξ ( i ) , for each i ∈ X m . 9
A General Version of G2 Theorem Feferman’s G2 plus Suppose U is consistent and ξ uniformly semi-numerates the Examples axioms X of U ( w.r.t. N ) . Then, U � con N [ ξ ] . Uniform Semi-numerability The Henkin The square brackets emphasise that ξ is not supposed to be Calculus relativised to N . Let B be a single sentence that axiomatises S 1 2 . The µ -Calculus Proof. Suppose U ⊢ con N [ ξ ] . Then, for some k , U k ⊢ B N ∧ con N [ ξ ] and ξ semi-numerates X k in U k . Let β := � A ∈ X k ( x = � A � ) . We find U k ⊢ ( B ∧ con ( β )) N . This contradicts a standard version of G2. ❑ prov N [ ξ ] need not to satisfy the Löb Conditions. Yet the Löb Condtions are used in the proof. 10
Löb’s Rule Feferman’s G2 plus Examples Uniform Semi-numerability Since, uniformity can be easily lifted to finite extensions, we have: The Henkin Calculus The µ -Calculus Theorem Suppose ξ uniformly semi-numerates the axioms of U ( w.r.t. N ) and U ⊢ prov N [ ξ ] ( � A � ) → A. Then U ⊢ A. 11
Overview Feferman’s G2 plus Examples Feferman’s G2 plus Examples Uniform Semi-numerability The Henkin Calculus The µ -Calculus Uniform Semi-numerability The Henkin Calculus The µ -Calculus 12
The Henkin Calculus Feferman’s G2 plus Examples We work in the language of modal logic extended with a fixed-point Uniform Semi-numerability operator ̥ p . ϕ . HC has the following axioms and rules. The Henkin Calculus ◮ The axioms and rules of K, The µ -Calculus ◮ Löb’ rule: if ⊢ ϕ → ϕ , then ⊢ ϕ . ◮ If ψ results from ϕ by renaming bound variables, then ⊢ ϕ ↔ ψ . ◮ If ̥ p . ϕ is substitutable for p in ϕ , then ϕ [ p : ̥ p . ϕ ] . ⊢ ̥ p . ϕ ↔ 13
Perspective The Henkin Calculus has standard syntax here. The disadvantage Feferman’s G2 plus Examples is that one has to get the details for variable-binding right —as is Uniform Semi-numerability witnessed by the presence of the α -equivalence axiom. The Henkin Calculus One gets a sense that this material is about circularity rather than The µ -Calculus binding. A treatment using syntax on possibly cyclic graphs seems to represent what is going on much better. Such a treatment would be co-inductive. The disadvantage is discontinuity with conventional treatments. The disadvantage of the second approach can, hopefully, be overcome by metatheorems linking the conventional treatment to the co-inductive one. 14
Circular Henkin Calculus Feferman’s G2 plus Examples This is what the Henkin Calculus looks like on directed possibly Uniform circular graphs. Note that ̥ is not in the language here. Semi-numerability The Henkin Calculus We demand that in a graph that represents a formula every cycle The µ -Calculus contains a vertex labeled with a box. This is the guarding condition. ◮ The Axioms and Rules of K. ◮ Löb’s rule: if ⊢ ϕ → ϕ , then ⊢ ϕ . ◮ If ϕ and ψ are bisimilar then ⊢ ϕ ↔ ψ . 15
The Grullet Modality Back to the ordinary syntax. Feferman’s G2 plus Examples We define: Uniform Semi-numerability • ϕ := ̥ p . ( ϕ ∧ p ) , where p is not free in ϕ . ◮ The Henkin Calculus The µ -Calculus We have: • ϕ ↔ • ϕ ) . ◮ HC ⊢ ( ϕ ∧ ◮ HC verifies Löb’s Logic for • . ◮ Suppose ϕ is modalised in p , then HC ⊢ ( � • ( p ↔ ϕ ) ∧ � • ( q ↔ ϕ [ p : q ])) → ( p ↔ q ) . A version of the De Jongh-Sambin-Bernardi Theorem Gödel and Henkin sentences are unique. 16
Multiple Fixed Points 1 Feferman’s G2 plus Consider a system of equations E given by: Examples Uniform Semi-numerability ( p 0 ↔ ϕ 0 ) , . . . , ( p n − 1 ↔ ϕ n − 1 ) , The Henkin Calculus We assign a directed graph G E to E with as nodes the variables p i , The µ -Calculus for i < n . We have an arrow from p i to p j iff there is an unguarded free occurrence of p j in ϕ i . E is guarded iff G E is acyclic. If E is guarded, it has a unique solution. In this solution the free variables are those of the ϕ i minus the p j . 17
Multiple Fixed Points 2 Feferman’s G2 plus Examples Uniform Semi-numerability The Henkin Calculus The µ -Calculus Figure: The associated graph 18
Reduction Feferman’s G2 plus Examples Suppose ϕ is modalised in p . We can find a � ϕ and fresh Uniform Semi-numerability q 0 , . . . , q n − 1 , such that p does not occur in � ϕ and The Henkin Calculus HC ⊢ ϕ ↔ � ϕ [ q 0 : ψ 0 , . . . , q n − 1 : ψ n − 1 ] . The µ -Calculus By solving the equations E : p ↔ � ϕ, q 0 ↔ ψ 0 , . . . , q n − 1 ↔ ψ n − 1 . we see that ϕ has a unique fixed point. 19
The Extended Henkin Calculus Feferman’s G2 plus Examples Using the reduction result we can show that the Henkin Calculus Uniform Semi-numerability is synonymous with its extended variant where we have fixed The Henkin Calculus points for modalised formulas: The µ -Calculus ◮ we have ̥ p .ϕ in case p only occurs in the scope of a box. The axioms for the extended calculus are analogous. On the circular syntax the difference between both versions disappears. 20
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