G odels Theorem: Inconsistency vs Introduction: Incompleteness - PowerPoint PPT Presentation

G¨ odel’s Theorem: Inconsistency vs Incomplete- ness Graham Priest G¨ odel’s Theorem: Inconsistency vs Introduction: Incompleteness the Standard View G¨ odel’s Proof The Graham Priest Inconsistency of Arithmetic Non-Triviality Naive Arithmetic November 4, 2016 and Axiomati- zability Coda: G¨ odel’s Second Incom- pleteness Theorem Graham Priest G¨ odel’s Theorem: Inconsistency vs Incompleteness

Plan G¨ odel’s Theorem: Inconsistency vs Incomplete- 1 Introduction: the Standard View ness Graham Priest 2 G¨ odel’s Proof Introduction: the Standard View 3 The Inconsistency of Arithmetic G¨ odel’s Proof The Inconsistency 4 Non-Triviality of Arithmetic Non-Triviality 5 Naive Arithmetic and Axiomatizability Naive Arithmetic and Axiomati- zability 6 Coda: G¨ odel’s Second Incompleteness Theorem Coda: G¨ odel’s Second Incom- pleteness Theorem Graham Priest G¨ odel’s Theorem: Inconsistency vs Incompleteness

Statement of the Theorem G¨ odel’s Theorem: Inconsistency vs Incomplete- ness Graham Priest G¨ odel’s first incompleteness theorem : any axiomatic Introduction: the Standard theory of arithmetic, with appropriate expressive View capabilities, is incomplete. G¨ odel’s Proof The Inconsistency of Arithmetic Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨ odel’s Second Incom- pleteness Theorem Graham Priest G¨ odel’s Theorem: Inconsistency vs Incompleteness

Statement of the Theorem G¨ odel’s Theorem: Inconsistency vs Incomplete- ness Graham Priest G¨ odel’s first incompleteness theorem : any axiomatic Introduction: the Standard theory of arithmetic, with appropriate expressive View capabilities, is incomplete. G¨ odel’s Proof The Inconsistency of Arithmetic Inaccurate : it must be either incomplete or inconsistent. Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨ odel’s Second Incom- pleteness Theorem Graham Priest G¨ odel’s Theorem: Inconsistency vs Incompleteness

Assumptions about T G¨ odel’s Theorem: Inconsistency vs Incomplete- ness A G¨ odel codes are assigned to syntactic entities, such as Graham Priest formulas and proofs. If n is a number, write its numeral as Introduction: n . If A is a formula with code n , write � A � for n . the Standard View G¨ odel’s Proof The Inconsistency of Arithmetic Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨ odel’s Second Incom- pleteness Theorem Graham Priest G¨ odel’s Theorem: Inconsistency vs Incompleteness

Assumptions about T G¨ odel’s Theorem: Inconsistency vs Incomplete- ness A G¨ odel codes are assigned to syntactic entities, such as Graham Priest formulas and proofs. If n is a number, write its numeral as Introduction: n . If A is a formula with code n , write � A � for n . the Standard View G¨ odel’s Proof B There is a formula with two free variables, B ( x , y ), which The represents the proof relation of T . That is: Inconsistency of Arithmetic (i) if n is the code of a proof of A in T then B ( n , � A � ) is true Non-Triviality in the standard model Naive (ii) if n is the not code of a proof of A in T then ¬ B ( n , � A � ) is Arithmetic and Axiomati- true in the standard model zability Coda: G¨ odel’s Second Incom- pleteness Theorem Graham Priest G¨ odel’s Theorem: Inconsistency vs Incompleteness

Assumptions Ctd. G¨ odel’s Theorem: Inconsistency vs Incomplete- ness Graham Priest C Define Prov ( y ) as ∃ xB ( x , y ). Then Prov is a proof Introduction: the Standard predicate for T . That is: View if T ⊢ A then T ⊢ Prov � A � G¨ odel’s Proof The Inconsistency of Arithmetic Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨ odel’s Second Incom- pleteness Theorem Graham Priest G¨ odel’s Theorem: Inconsistency vs Incompleteness

Assumptions Ctd. G¨ odel’s Theorem: Inconsistency vs Incomplete- ness Graham Priest C Define Prov ( y ) as ∃ xB ( x , y ). Then Prov is a proof Introduction: the Standard predicate for T . That is: View if T ⊢ A then T ⊢ Prov � A � G¨ odel’s Proof The Inconsistency of Arithmetic D There is a formula, G , of the form ¬ Prov � G � Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨ odel’s Second Incom- pleteness Theorem Graham Priest G¨ odel’s Theorem: Inconsistency vs Incompleteness

Proof G¨ odel’s If T ⊢ G then T ⊢ ¬ Prov � G � . Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨ odel’s Proof The Inconsistency of Arithmetic Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨ odel’s Second Incom- pleteness Theorem Graham Priest G¨ odel’s Theorem: Inconsistency vs Incompleteness

Proof G¨ odel’s If T ⊢ G then T ⊢ ¬ Prov � G � . Theorem: Inconsistency If T ⊢ G then T ⊢ Prov � G � . vs Incomplete- ness Graham Priest Introduction: the Standard View G¨ odel’s Proof The Inconsistency of Arithmetic Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨ odel’s Second Incom- pleteness Theorem Graham Priest G¨ odel’s Theorem: Inconsistency vs Incompleteness

Proof G¨ odel’s If T ⊢ G then T ⊢ ¬ Prov � G � . Theorem: Inconsistency If T ⊢ G then T ⊢ Prov � G � . vs Incomplete- ness So if T ⊢ G , T is inconsistent. Graham Priest Introduction: the Standard View G¨ odel’s Proof The Inconsistency of Arithmetic Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨ odel’s Second Incom- pleteness Theorem Graham Priest G¨ odel’s Theorem: Inconsistency vs Incompleteness

Proof G¨ odel’s If T ⊢ G then T ⊢ ¬ Prov � G � . Theorem: Inconsistency If T ⊢ G then T ⊢ Prov � G � . vs Incomplete- ness So if T ⊢ G , T is inconsistent. Graham Priest Suppose that T is consistent. Introduction: the Standard View G¨ odel’s Proof The Inconsistency of Arithmetic Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨ odel’s Second Incom- pleteness Theorem Graham Priest G¨ odel’s Theorem: Inconsistency vs Incompleteness

Proof G¨ odel’s If T ⊢ G then T ⊢ ¬ Prov � G � . Theorem: Inconsistency If T ⊢ G then T ⊢ Prov � G � . vs Incomplete- ness So if T ⊢ G , T is inconsistent. Graham Priest Suppose that T is consistent. Introduction: T �⊢ G the Standard View G¨ odel’s Proof The Inconsistency of Arithmetic Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨ odel’s Second Incom- pleteness Theorem Graham Priest G¨ odel’s Theorem: Inconsistency vs Incompleteness

Proof G¨ odel’s If T ⊢ G then T ⊢ ¬ Prov � G � . Theorem: Inconsistency If T ⊢ G then T ⊢ Prov � G � . vs Incomplete- ness So if T ⊢ G , T is inconsistent. Graham Priest Suppose that T is consistent. Introduction: T �⊢ G the Standard View No number is the code of a proof of G . G¨ odel’s Proof The Inconsistency of Arithmetic Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨ odel’s Second Incom- pleteness Theorem Graham Priest G¨ odel’s Theorem: Inconsistency vs Incompleteness

Proof G¨ odel’s If T ⊢ G then T ⊢ ¬ Prov � G � . Theorem: Inconsistency If T ⊢ G then T ⊢ Prov � G � . vs Incomplete- ness So if T ⊢ G , T is inconsistent. Graham Priest Suppose that T is consistent. Introduction: T �⊢ G the Standard View No number is the code of a proof of G . G¨ odel’s Proof For any n , ¬ B ( n , � G � ) is true in the standard model. The Inconsistency of Arithmetic Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨ odel’s Second Incom- pleteness Theorem Graham Priest G¨ odel’s Theorem: Inconsistency vs Incompleteness

Proof G¨ odel’s If T ⊢ G then T ⊢ ¬ Prov � G � . Theorem: Inconsistency If T ⊢ G then T ⊢ Prov � G � . vs Incomplete- ness So if T ⊢ G , T is inconsistent. Graham Priest Suppose that T is consistent. Introduction: T �⊢ G the Standard View No number is the code of a proof of G . G¨ odel’s Proof For any n , ¬ B ( n , � G � ) is true in the standard model. The Inconsistency ∀ x ¬ B ( x , � G � ) is true in the standard model of Arithmetic Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨ odel’s Second Incom- pleteness Theorem Graham Priest G¨ odel’s Theorem: Inconsistency vs Incompleteness

Proof G¨ odel’s If T ⊢ G then T ⊢ ¬ Prov � G � . Theorem: Inconsistency If T ⊢ G then T ⊢ Prov � G � . vs Incomplete- ness So if T ⊢ G , T is inconsistent. Graham Priest Suppose that T is consistent. Introduction: T �⊢ G the Standard View No number is the code of a proof of G . G¨ odel’s Proof For any n , ¬ B ( n , � G � ) is true in the standard model. The Inconsistency ∀ x ¬ B ( x , � G � ) is true in the standard model of Arithmetic ¬∃ xB ( x , � G � ) Non-Triviality ¬ Prov � G � Naive Arithmetic G and Axiomati- zability G is true in the standard model. Coda: G¨ odel’s Second Incom- pleteness Theorem Graham Priest G¨ odel’s Theorem: Inconsistency vs Incompleteness