The logic of Σ formulas

Transcription

1 The logic of Σ formulas Andre Kornell UC Davis BLAST August 10, 2018 Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, / 22

2 the Vienna Circle The meaning of a proposition is the method of its verification. - Moritz Schlick, in Meaning and Verification Say: a proposition is verifiable iff it admits a method of verification. The negation of a verifiable proposition need not be verifiable. examples (ignoring experimental error) 1 a 3 + b 3 = c 3 is solvable vs a 3 + b 3 = c 3 is not solvable 2 ZFC is inconsistent vs ZFC is consistent 3 α 1/137 vs α = 1/137 Verifiable propositions are closed under conjunction and disjunction. Verifiable propositions are not closed under negation. Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, / 22

3 propositional positivistic logic connectives definition A positivistic propositional theory T consists of implications α = β, with α and β positivistic. A valuation m models T just in case m = α implies m = β, for all α = β in T. completeness theorem Let T be a positivistic propositional theory, and let φ = ψ be an implication between positivistic formulas. Then, T proves φ ψ if and only if every valuation that models T also models φ ψ. Proof systems: Hilbert style, sequent calculus, etc. Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, / 22

4 meaningful deduction For φ and ψ positivistic formulas in verifiable propositional constants: φ and ψ are always both verifiable φ ψ is generally not verifiable A deduction establishes nothing unless it consists of meaningful formulas. To a finitist: meaningful means verifiable by finite computation To a realist: meaningful means verifiable by transfinite computation (Tarski s definition of truth.) An axiom α = β of should be interpreted as a rule of inference. Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, / 22

5 deep inference Example. propositional constants P, Q 1, Q 2, R. theory T = {P Q 1 Q 2, Q 1 R, Q 2 }. A deduction from P To R: P Q 1 Q 2 R Q 2 R R R R We define the deductive system RK(T ). For each implication α β that is a logical axiom or an axiom of T, we have the rule: Φ(α) Φ(β) Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, / 22

6 logical axioms of propositional positivistic logic ψ φ φ ψ φ φ ψ ψ φ φ φ ψ ψ ψ φ φ ψ ψ φ ψ (φ ψ) χ (φ χ) (ψ χ) Theorem (K) For positivistic propositional theories T, and implications φ ψ, TFAE: 1 every valuation modeling T models φ ψ 2 there is a classical derivation of T φ ψ 3 there is an intuisionistic derivation of T φ ψ 4 there is a deduction of φ ψ in RK(T ) 5 every bounded distributive lattice modeling T models φ ψ Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, / 22

7 coherent logic A coherent theory T is consist of implications between coherent formulas. φ(t, w) = v : φ(v, w) v : ψ(w) = ψ(w) φ(w) v : ψ(v, w) = v : φ(w) ψ(v, w) We define the deductive system RK (T ). For each implication α β that is a logical axiom or an axiom of T, we have the rule: Φ(α(t 1,..., t n )) Φ(β(t 1,..., t n )) Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, / 22

8 completeness for coherent logic Theorem (K) For coherent theories T, and implications φ ψ, TFAE: 1 every model of T models φ ψ 2 there is a classical derivation of T φ ψ 3 there is an intuisionistic derivation of T φ ψ 4 there is a deduction of φ ψ in RK (T ) The free variables in a deduction can be treated as constant symbols. Compare to the induction rule of primitive recursive arithmetic: φ(0) φ(v, w) φ(s(v), w) φ(v, w) Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, / 22

9 completed, surveyable, definite Universal quantification is suspect whenever we have an infinite universe. potential infinity vs. completed infinity Even if the natural numbers form a completed totality, the totality of all sets need not be. (Russell s paradox.) In the computational framework: A class is surveyable just in case there is a (transfinite) process that surveys, i. e., sorts through the totality. (Weaver) Similarly, a class is definite just in case quantification takes bivalent propositions to bivalent proposition. (Feferman) completed surveyable definite realist universal quantification Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, / 22

10 universal quantification ψ(w) = v : ψ(w) v : φ(v, w) = φ(t, w) We define the system RI, analogously to RK. Theorem (K) Let T be a theory in coherent logic with universal quantification. For each implication φ ψ, TFAE: 1 There is an intuitionistic derivation of T φ ψ 2 there is a deduction of φ ψ in RI, (T ) We do not have a completeness result for RI,! Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, / 22

11 surveyability axiom We express realist universal quantification: v : φ(w) ψ(v, w) = φ(w) v : ψ(v, w) If we can survey the universe, then we can either verify φ(w) or verify ψ(v, w) for each value of v. We define the deductive system RK, (T ) to be the system RI, (T ) together with the above schema of logical axioms. Theorem (K) Let T be a theory in coherent logic with universal quantification. For each implication φ ψ, TFAE: 1 every model of T models φ ψ 2 there is a classical derivation of T φ ψ 3 there is a deduction of φ ψ in RK, (T ) Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, / 22

12 negation In applications, our primitive predicates are usually decidable. P(w) P(w) P(w) P(w) In this case, RK, (T ) is essentially a classical system. classical logic decidable primitive predicates surveyable universe In applications, the universe is usually not surveyable/definite/completed, but the universe is rather the union of surveyable subclasses. (sets) v : v t : Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, / 22

13 positivistic logic definition The class of Σ formulas is the closure of the atomic formulas for the forms φ ψ φ ψ v : φ v t : ψ The system RK Σ (T ) has the axiom schemes as RK (T ) together with the following logical axioms: x y x y x y y x ψ(w) = v t : ψ(w) v t : φ(v, w) = φ(s, w) v t : φ(w) ψ(v, w) = φ(w) v t : ψ(v, w) v t : v t φ(v, w) = v t : φ(v, w) Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, / 22

14 completeness Theorem (K) Let T be a set of implications between Σ formulas, and let φ and ψ be Σ formulas. TFAE: 1 every model of T is a model of φ ψ 2 there is a classical deduction of T φ ψ. 3 there is a deduction of φ ψ in RK Σ (T ). The equivalence (2) (3) is a theorem of IΣ 1. Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, / 22

15 equality In applications, equality is usually available. 1 = x = x 2 x = y = y = x 3 x = y y = z = x = z 4 φ(x, w) x = y = φ(y, w) 5 = x = y x y 6 x = y y x = Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, / 22

16 set theory T 1 1 ( z x : z y) ( z y : z x) = x = y 2 z {x, y} z = x z = y 3 z x y x : z x 4 y (x) z y : z x 5 Y (X ): x X : x Y φ 6 z x : = z x : y x : z y 7 x X : y : φ = Y : x X : y Y : φ 8 = x : Ord(x) x z 9 = V : Mod(V ) z V Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, / 22

17 subuniverses Mod(V ) abbreviates: ( x V : y x : y V ) ( x V : y V : {x, y} V ) ( x V : x V ) ( x V : (x) V ) ( x V : y (V ): x y y V ) corollary* The deductive system RK Σ (T 1 ) deduces Mod(V ) φ V for every axiom φ of ZFC. Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, / 22

18 the truth predicate Well-known Σ truth predicate for Σ formulas: T(φ ψ) w : Wit(w, φ) 1 T(φ ψ) T(φ) T(ψ) 2 T(φ ψ) T(φ) T(ψ) 3 T( v : ψ(v)) a: T(ψ(a)) 4 T( v b : φ(v)) a b : T(φ(a)) 5 T(P(t 1,..., t n )) a 1 : a n : P(a 1,..., a n ) T(a 1 = t 1 )... T(a n = t n ) 6 T(a = F (t 1,..., t n )) b 1 : b n : a = F (b 1,..., b n ) T(b 1 = t 1 )... T(b n = t n ) Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, / 22

19 metamathematics is mathematics Define the theory T 2 to be the theory T 1 together with the axiom T(φ) I T1 (φ ψ) = T(ψ) The predicate I T (φ ψ) expresses that φ ψ is a correct elementary inference of the rewriting system RK Σ (T ). (a single deductive step) proposition (K) The theory T 2 is finitely axiomatizable, and RK Σ (T 2 ) deduces T(φ) I T2 (φ ψ) = T(ψ). Thus, T 2 describes a self-contained incompletable universe of pure sets, that includes nested transitive models of ZFC. The object language and the metalanguage coincide. Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, / 22

20 primitive recursive set theory The primitive recursive set functions (Jensen and Karp) are obtained using the following recursion scheme: ( ) F (v, w) = H {F (r, w) r v}, v, w Rathjen defined a theory PRS of primitive recursive set functions, analogous to Skolem s theory PRA of primitive recursive functions. Both PRS and PRA can be formulated as positivistic theories. Define a positivistic theory PRS + by adding to PRS the Σ-collection schema, and the well-ordering principle. PRS + is essentially a weak fragment of KP + WOP. It is strong enough to formalize elementary constructions. Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, / 22

21 the system LK ω Introduction Rules for and ( φ K) Γ, φ Γ, K ( φ K) Γ, φ Γ, K ( φ K) Γ, φ Γ, K ( φ K) Γ, φ Γ, K Cut Rule Γ, φ Γ, φ Γ, Γ, Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, / 22

22 model completeness principle Let T be a set of LK ω formulas. Conclude that T is inconsistent for LK ω or that T has a model. set completeness principle Let A be a transitive set. Let T be a set of LK ω formulas in the vocabulary {=,, S} and parameters from A. Assume that T includes basic axioms describing the structure (A, =, ). Conclude that T is inconsistent for LK ω or that there exists a set B A such that (A, =,, B) is a model of T. theorem (K) Work in PRS + together with cut-elimination* for LK ω. Each may be proved using the other as an axiom: 1 the model completeness principle 2 the set completeness principle together with the axiom of infinity Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, / 22