Modal Quantifiers, Potential Infinity, and Yablo sequences Rafa - PowerPoint PPT Presentation

Modal Quantifiers, Potential Infinity, and Yablo sequences Rafał Urbaniak (Ghent U., U. of Gda ń sk) Michał T. Godziszewski (U. of Warsaw) [stu ff on Yablo sequences] ICLA, Delhi 2019 1/ 28

Yablo’s paradox Arithmetization of Yablo sentences Potentially infinite domains and sl-semantics Modal interpretation of quantifiers Yablo sequences with modal quantifiers Summing up 2/ 28

Yablo’s paradox Y 0 For any k > 0, Y k is false. Y 1 For any k > 1, Y k is false. Y 2 For any k > 2, Y k is false. . . . Y n For any k > n , Y k is false. . . . 3/ 28

Yablo’s paradox Y 0 For any k > 0, Y k is false. Y 1 For any k > 1, Y k is false. Y 2 For any k > 2, Y k is false. . . . Y n For any k > n , Y k is false. . . . Suppose Y n . So for any j > n , ¬ Y j . So ¬ Y n +1 and for any j > n + 1, ¬ Y j . So Y n +1 . Contradiction. So ¬ Y n unconditionally. So ∃ k > n Y k . Rinse and repeat. 3/ 28

Background assumptions (Ketland, 2005) Uniform disquotation ∀ x ( Y ( x ) ≡ Tr ( Y ( x ))) Local disquotation For any particular n , assume Y (¯ n ) ≡ Tr ( Y (¯ n )). ω -rule If for any n ϕ (¯ n ), derive ∀ x ϕ ( x ). 4/ 28

Finitistic way out? The idea If the world is finite, there are only finitely many Yablo sentences, and the last one is vacuously true. The challenge Make sense of arithmetic in a formal finitistic setting. The strategy There could be more things: potential infinity . 5/ 28

Existence of Yablo Formulas (Priest, 1997) Definition (Yablo formula) Y ( x ) is a Yablo formula in T i ff T ⊢ ∀ x ( Y ( x ) ≡ ∀ w > x ¬ Tr ( � Y ( ˙ w ) � )) . Yablo sentences are of the form Y (¯ n ). Theorem If T is nice, there exists a Yablo formula in T . 6/ 28

Existence of Yablo Formulas (Priest, 1997) Definition (Yablo formula) Y ( x ) is a Yablo formula in T i ff T ⊢ ∀ x ( Y ( x ) ≡ ∀ w > x ¬ Tr ( � Y ( ˙ w ) � )) . Yablo sentences are of the form Y (¯ n ). Theorem If T is nice, there exists a Yablo formula in T . Proof. Let ϕ ( x , y ) = ∀ w > x ¬ Tr ( sub ( y , � y � , name ( w ))). By the Diagonal Lemma, there is a formula Y ( x ) s.t.: T ⊢ Y ( x ) ≡ ∀ w > x ¬ Tr ( sub ( � Y ( x ) � , � y � , name ( w ))) . T ⊢ Y ( x ) ≡ ∀ w > x ¬ Tr ( � Y ( ˙ w ) � ). 6/ 28

ω -inconsistency of Yablo formulas (Ketland, 2005) Definition ( ω -consistency) T is ω -consistent i ff there is no ϕ ( x ) s.t. simultaneously: ∀ n ∈ ω T ⊢ ¬ ϕ ( n ) T ⊢ ∃ x ϕ ( x ) T is ω -inconsistent o/w. 7/ 28

ω -inconsistency of Yablo formulas (Ketland, 2005) Definition ( ω -consistency) T is ω -consistent i ff there is no ϕ ( x ) s.t. simultaneously: ∀ n ∈ ω T ⊢ ¬ ϕ ( n ) T ⊢ ∃ x ϕ ( x ) T is ω -inconsistent o/w. Definition ( PA F ) Let L F be standard language extended with F . PA F := PA ∪ { F ( n ) ≡ ∀ x > n ¬ F ( x ) : n ∈ ω } 7/ 28

ω -inconsistency of Yablo formulas (Ketland, 2005) Definition ( ω -consistency) T is ω -consistent i ff there is no ϕ ( x ) s.t. simultaneously: ∀ n ∈ ω T ⊢ ¬ ϕ ( n ) T ⊢ ∃ x ϕ ( x ) T is ω -inconsistent o/w. Definition ( PA F ) Let L F be standard language extended with F . PA F := PA ∪ { F ( n ) ≡ ∀ x > n ¬ F ( x ) : n ∈ ω } Theorem PA F is ω -inconsistent. 7/ 28

ω -inconsistency of Yablo formulas (Ketland, 2005) PA F := PA ∪ { F ( n ) ≡ ∀ x > n ¬ F ( x ) : n ∈ ω } Work in PA F . Fix an n ∈ ω and assume F ( n ). ∀ x > n ¬ F ( x ) . ( ⋆ ) 8/ 28

ω -inconsistency of Yablo formulas (Ketland, 2005) PA F := PA ∪ { F ( n ) ≡ ∀ x > n ¬ F ( x ) : n ∈ ω } Work in PA F . Fix an n ∈ ω and assume F ( n ). ∀ x > n ¬ F ( x ) . ( ⋆ ) In particular, ∀ x > n + 1 ¬ F ( x ). This is equivalent to F ( n + 1). But from ( ⋆ ), ¬ F ( n + 1) follows. Contradiction. So unconditionally ¬ F ( n ): ∀ n ∈ ω PA F ⊢ ¬ F ( n ) . (1) By definition of PA F : ∀ n ∈ ω PA F ⊢ ∃ x > n F ( x ) . In particular: PA F ⊢ ∃ x F ( x ) . (2) 8/ 28

The consistency of Yablo formulas Theorem PA F is consistent. 9/ 28

The consistency of Yablo formulas Theorem PA F is consistent. Proof. Take a nonstandard model M of PA . Pick a nonstandard a ∈ M , let A = { a } . Put F M = A . ∀ n ∈ ω ( M , A ) | = ¬ F ( n ) . But also, ( M , A ) | = ∃ x F ( x ). Moreover, ∀ n ∈ ω ( M , A ) | = ∃ x > n F ( x ). Hence ∀ n ∈ ω ( M , A ) | = F ( n ) ≡ ∀ x > n ¬ F ( x ) (both sides are false). So ( M , A ) | = PA F and PA F is consistent. 9/ 28

Adding local disquotation Definition AD = { Tr ( � ϕ � ) ≡ ϕ : ϕ ∈ Sent L } YD = { Tr ( � Y ( n ) � ) ≡ Y ( n ) : Y ( n ) belongs to the Yablo sequence } 10/ 28

Adding local disquotation Definition AD = { Tr ( � ϕ � ) ≡ ϕ : ϕ ∈ Sent L } YD = { Tr ( � Y ( n ) � ) ≡ Y ( n ) : Y ( n ) belongs to the Yablo sequence } Definition ( PA D ) PAT is obtained from PA by adding Tr . (induction!) PA D = PAT ∪ AD ∪ YD . PA − D is PA D with induction without Tr . 10/ 28

Adding local disquotation Theorem PA D is ω -inconsistent. Proof. Existence of YF entails: ∀ n ∈ ω PA D ⊢ Y ( n ) ≡ ∀ x > n ¬ Tr ( � Y ( ˙ x ) � ) . By the inclusion of YD we get: ∀ n ∈ ω PA D ⊢ Tr ( � Y ( n ) � ) ≡ ∀ x > n ¬ Tr ( � Y ( ˙ x ) � ) . Let F ( x ) := Tr ( � Y ( ˙ x ) � ): ∀ n ∈ ω PA D ⊢ F ( n ) ≡ ∀ x > n ¬ F ( x ) . So PA D contains PA F (which is ω -inconsistent). 11/ 28

The consistency of PA − D Theorem PA − D is consistent. Proof. Take a nonstandard M of PA . Let t ( x ) := � Y ( ˙ x ) � . By overspill, there are nonstandard b and c such that t M ( b ) = c . Let Tr M = S = Th L ( M ) ∪ { c } . Clearly, ( M , S ) | = AD . ∀ n ∈ ω ( M , S ) | = ∃ x > n Tr ( � Y ( ˙ x ) � ) ∀ n ∈ ω ( M , S ) | = ¬ Y (¯ n ) Standard Y ( n ) are not in S , so: ∀ n ∈ ω ( M , S ) | = ¬ Tr ( � Y (¯ n ) � ) . So ( M , S ) | = YD (UYD fails here). 12/ 28

The consistency of PA D Theorem PA D is consistent. Proof. By finite satisfiability (put only the last Yablo sentence in the extension of Tr , check induction holds), and compactness. 13/ 28

Conservativeness of PA D Theorem PA D is a conservative extension of PA . 14/ 28

Conservativeness of PA D Theorem PA D is a conservative extension of PA . Proof. Suppose PA � ϕ . So PA ∪ {¬ ϕ } is consistent. For a nonstandard M of PA , M | = ¬ ϕ . There is an elementarily equivalent M ′ such that ( M ′ , Tr M ′ ) | = PA D . ( M ′ , Tr M ′ ) �| = ϕ , and so PA D � ϕ . 14/ 28

Uniform Yablo Disquotation yields contradiction Definition UYD = ∀ x ( Tr ( � Y ( ˙ x ) � ) ≡ Y ( x )) 15/ 28

Uniform Yablo Disquotation yields contradiction Definition UYD = ∀ x ( Tr ( � Y ( ˙ x ) � ) ≡ Y ( x )) Theorem Let S = PAT + UYD . S is inconsistent. 15/ 28

Uniform Yablo Disquotation yields contradiction Definition UYD = ∀ x ( Tr ( � Y ( ˙ x ) � ) ≡ Y ( x )) Theorem Let S = PAT + UYD . S is inconsistent. Work in S . ∀ x ( Y ( x ) ≡ ∀ w > x ¬ Tr ( � Y ( ˙ w ) � )) [Yablo existence] UYD gives ∀ x ( Y ( x ) ≡ ∀ w > x ¬ Y ( w )). So ∀ x ( Y ( x ) ≡ ∀ w > x ∃ z > w Tr ( � Y ( ˙ z ) � )) [unraveling] By UYD: ∀ x ( Y ( x ) ≡ ∀ w > x ∃ z > w Y ( z )) So ∀ x ( Y ( x ) ≡ ∃ w > x Y ( w )) ∀ x (( ∀ w > x ¬ Y ( w )) ≡ ( ∃ w > xY ( w ))) 15/ 28

Local disquotation with ω -rule is inconsistent Theorem = ( PAT − ∪ AD ∪ YD ) ω . PA ω − Let PA ω − is inconsistent. D D ( AD is not needed.) 16/ 28

Local disquotation with ω -rule is inconsistent Theorem = ( PAT − ∪ AD ∪ YD ) ω . PA ω − Let PA ω − is inconsistent. D D ( AD is not needed.) Proof idea. ∀ n ∈ ω PA ω − ⊢ ¬ Y ( n ) [internalized standard reasoning] D ∀ n ∈ ω PA ω − ⊢ ¬ Tr ( � Y ( n ) � ) [Y disquotation] D PA ω − ⊢ ∀ x ¬ Tr ( � Y ( ˙ x ) � ) [ ω -rule] D In particular: PA ω − ⊢ Y (23) D 16/ 28

Classical set-up vs. Yablo Even those theories which prove the existence of Yablo sentences are still consistent. They’re ω -inconsistent with local Yablo disquotation, though. One way to obtain a contradiction: uniform Yablo disquotation. Another one: local disquotation and ω − rule . 17/ 28

sl-semantics (Mostowski, 2001a,b, 2016) Definition ( FM -domains) Take a relational arithmetical language. FM ( N ) = { N n : n = 1 , 2 ,... } N n = ( { 0 , 1 ,..., n − 1 } , + ( n ) , × ( n ) , 0 ( n ) , s ( n ) ,< ( n ) ) . 18/ 28

sl-semantics (Mostowski, 2001a,b, 2016) Definition ( FM -domains) Take a relational arithmetical language. FM ( N ) = { N n : n = 1 , 2 ,... } N n = ( { 0 , 1 ,..., n − 1 } , + ( n ) , × ( n ) , 0 ( n ) , s ( n ) ,< ( n ) ) . Definition ( sl -theory of FM ( N )) Satisfaction in finite points in FM ( N ) is standard. FM ( N ) | = sl ϕ i ff ∃ m ∀ k ( k ≥ m ⇒ N k | = ϕ ) sl ( FM ( N )) = { ϕ ∈ Sent L : FM ( N ) | = sl ϕ } 18/ 28