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

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

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G odels Theorem: Inconsistency vs Incomplete- ness Graham Priest G odels Theorem: Inconsistency vs Introduction: Incompleteness the Standard View G odels Proof The Graham Priest Inconsistency of Arithmetic

slide-1
SLIDE 1

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

G¨

  • del’s Theorem: Inconsistency vs

Incompleteness

Graham Priest November 4, 2016

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-2
SLIDE 2

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Plan

1 Introduction: the Standard View 2 G¨

  • del’s Proof

3 The Inconsistency of Arithmetic 4 Non-Triviality 5 Naive Arithmetic and Axiomatizability 6 Coda: G¨

  • del’s Second Incompleteness Theorem

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-3
SLIDE 3

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Statement of the Theorem

G¨

  • del’s first incompleteness theorem: any axiomatic

theory of arithmetic, with appropriate expressive capabilities, is incomplete.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-4
SLIDE 4

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Statement of the Theorem

G¨

  • del’s first incompleteness theorem: any axiomatic

theory of arithmetic, with appropriate expressive capabilities, is incomplete. Inaccurate: it must be either incomplete or inconsistent.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-5
SLIDE 5

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Assumptions about T

A G¨

  • del codes are assigned to syntactic entities, such as

formulas and proofs. If n is a number, write its numeral as

  • n. If A is a formula with code n, write A for n.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-6
SLIDE 6

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Assumptions about T

A G¨

  • del codes are assigned to syntactic entities, such as

formulas and proofs. If n is a number, write its numeral as

  • n. If A is a formula with code n, write A for n.

B There is a formula with two free variables, B(x, y), which represents the proof relation of T. That is:

(i) if n is the code of a proof of A in T then B(n, A) is true in the standard model (ii) if n is the not code of a proof of A in T then ¬B(n, A) is true in the standard model

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-7
SLIDE 7

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Assumptions Ctd.

C Define Prov(y) as ∃xB(x, y). Then Prov is a proof predicate for T. That is:

if T ⊢ A then T ⊢ Prov A

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-8
SLIDE 8

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Assumptions Ctd.

C Define Prov(y) as ∃xB(x, y). Then Prov is a proof predicate for T. That is:

if T ⊢ A then T ⊢ Prov A

D There is a formula, G, of the form ¬Prov G

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-9
SLIDE 9

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proof

If T ⊢ G then T ⊢ ¬Prov G.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-10
SLIDE 10

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proof

If T ⊢ G then T ⊢ ¬Prov G. If T ⊢ G then T ⊢ Prov G.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-11
SLIDE 11

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proof

If T ⊢ G then T ⊢ ¬Prov G. If T ⊢ G then T ⊢ Prov G. So if T ⊢ G, T is inconsistent.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-12
SLIDE 12

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proof

If T ⊢ G then T ⊢ ¬Prov G. If T ⊢ G then T ⊢ Prov G. So if T ⊢ G, T is inconsistent. Suppose that T is consistent.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-13
SLIDE 13

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proof

If T ⊢ G then T ⊢ ¬Prov G. If T ⊢ G then T ⊢ Prov G. So if T ⊢ G, T is inconsistent. Suppose that T is consistent. T ⊢ G

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-14
SLIDE 14

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proof

If T ⊢ G then T ⊢ ¬Prov G. If T ⊢ G then T ⊢ Prov G. So if T ⊢ G, T is inconsistent. Suppose that T is consistent. T ⊢ G No number is the code of a proof of G.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-15
SLIDE 15

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proof

If T ⊢ G then T ⊢ ¬Prov G. If T ⊢ G then T ⊢ Prov G. So if T ⊢ G, T is inconsistent. Suppose that T is consistent. T ⊢ G No number is the code of a proof of G. For any n, ¬B(n, G) is true in the standard model.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-16
SLIDE 16

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proof

If T ⊢ G then T ⊢ ¬Prov G. If T ⊢ G then T ⊢ Prov G. So if T ⊢ G, T is inconsistent. Suppose that T is consistent. T ⊢ G No number is the code of a proof of G. For any n, ¬B(n, G) is true in the standard model. ∀x¬B(x, G) is true in the standard model

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-17
SLIDE 17

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proof

If T ⊢ G then T ⊢ ¬Prov G. If T ⊢ G then T ⊢ Prov G. So if T ⊢ G, T is inconsistent. Suppose that T is consistent. T ⊢ G No number is the code of a proof of G. For any n, ¬B(n, G) is true in the standard model. ∀x¬B(x, G) is true in the standard model

¬∃xB(x, G) ¬Prov G G

G is true in the standard model.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-18
SLIDE 18

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proof

If T ⊢ G then T ⊢ ¬Prov G. If T ⊢ G then T ⊢ Prov G. So if T ⊢ G, T is inconsistent. Suppose that T is consistent. T ⊢ G No number is the code of a proof of G. For any n, ¬B(n, G) is true in the standard model. ∀x¬B(x, G) is true in the standard model

¬∃xB(x, G) ¬Prov G G

G is true in the standard model. So if T is consistent, it is incomplete. Contrapositively, if T is complete, it is inconsistent.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-19
SLIDE 19

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

L¨

  • b’s Principle

if Prov A then A

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-20
SLIDE 20

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Naive Arithmetic

Fix an appropriate language for first-order arithmetic.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-21
SLIDE 21

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Naive Arithmetic

Fix an appropriate language for first-order arithmetic. Let T be the theory containing all the things which are analytically true in this language.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-22
SLIDE 22

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Naive Arithmetic

Fix an appropriate language for first-order arithmetic. Let T be the theory containing all the things which are analytically true in this language. We do not assume that T is axiomatic.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-23
SLIDE 23

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Naive Arithmetic

Fix an appropriate language for first-order arithmetic. Let T be the theory containing all the things which are analytically true in this language. We do not assume that T is axiomatic. Write ⊢ for provability in T.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-24
SLIDE 24

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Assumptions

The basic facts about G¨

  • del codes can be established in

T.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-25
SLIDE 25

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Assumptions

The basic facts about G¨

  • del codes can be established in

T. The language contains a monadic predicate, P, which expresses this notion of provability in T.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-26
SLIDE 26

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Assumptions

The basic facts about G¨

  • del codes can be established in

T. The language contains a monadic predicate, P, which expresses this notion of provability in T.

[1] ⊢ ¬P A ∨ A [2] ⊢ A then ⊢ P A

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-27
SLIDE 27

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

T is Inconsistent

G is ¬P G

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-28
SLIDE 28

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

T is Inconsistent

G is ¬P G ⊢ ¬P G ∨ G

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-29
SLIDE 29

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

T is Inconsistent

G is ¬P G ⊢ ¬P G ∨ G ⊢ ¬P G ∨ ¬P G

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-30
SLIDE 30

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

T is Inconsistent

G is ¬P G ⊢ ¬P G ∨ G ⊢ ¬P G ∨ ¬P G ⊢ ¬P G

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-31
SLIDE 31

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

T is Inconsistent

G is ¬P G ⊢ ¬P G ∨ G ⊢ ¬P G ∨ ¬P G ⊢ ¬P G ⊢ G

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-32
SLIDE 32

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

T is Inconsistent

G is ¬P G ⊢ ¬P G ∨ G ⊢ ¬P G ∨ ¬P G ⊢ ¬P G ⊢ G ⊢ P G

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-33
SLIDE 33

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

T is Inconsistent

G is ¬P G ⊢ ¬P G ∨ G ⊢ ¬P G ∨ ¬P G ⊢ ¬P G ⊢ G ⊢ P G Note: This does not show that the P-free fragment of T is inconsistent.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-34
SLIDE 34

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proof of L¨

  • bs Theorem

Let A be any sentence. L is Prov L ⊃ A.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-35
SLIDE 35

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proof of L¨

  • bs Theorem

Let A be any sentence. L is Prov L ⊃ A. T ⊢ L ⊃ (Prov L ⊃ A)

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-36
SLIDE 36

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proof of L¨

  • bs Theorem

Let A be any sentence. L is Prov L ⊃ A. T ⊢ L ⊃ (Prov L ⊃ A) T ⊢ Prov L ⊃ (Prov L ⊃ A)

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-37
SLIDE 37

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proof of L¨

  • bs Theorem

Let A be any sentence. L is Prov L ⊃ A. T ⊢ L ⊃ (Prov L ⊃ A) T ⊢ Prov L ⊃ (Prov L ⊃ A) T ⊢ Prov L ⊃ Prov Prov L ⊃ A

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-38
SLIDE 38

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proof of L¨

  • bs Theorem

Let A be any sentence. L is Prov L ⊃ A. T ⊢ L ⊃ (Prov L ⊃ A) T ⊢ Prov L ⊃ (Prov L ⊃ A) T ⊢ Prov L ⊃ Prov Prov L ⊃ A T ⊢ Prov L ⊃ (Prov L ⊃ A)

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-39
SLIDE 39

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proof of L¨

  • bs Theorem

Let A be any sentence. L is Prov L ⊃ A. T ⊢ L ⊃ (Prov L ⊃ A) T ⊢ Prov L ⊃ (Prov L ⊃ A) T ⊢ Prov L ⊃ Prov Prov L ⊃ A T ⊢ Prov L ⊃ (Prov L ⊃ A) T ⊢ Prov L ⊃ A

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-40
SLIDE 40

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proof of L¨

  • bs Theorem

Let A be any sentence. L is Prov L ⊃ A. T ⊢ L ⊃ (Prov L ⊃ A) T ⊢ Prov L ⊃ (Prov L ⊃ A) T ⊢ Prov L ⊃ Prov Prov L ⊃ A T ⊢ Prov L ⊃ (Prov L ⊃ A) T ⊢ Prov L ⊃ A T ⊢ L

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-41
SLIDE 41

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proof of L¨

  • bs Theorem

Let A be any sentence. L is Prov L ⊃ A. T ⊢ L ⊃ (Prov L ⊃ A) T ⊢ Prov L ⊃ (Prov L ⊃ A) T ⊢ Prov L ⊃ Prov Prov L ⊃ A T ⊢ Prov L ⊃ (Prov L ⊃ A) T ⊢ Prov L ⊃ A T ⊢ L T ⊢ Prov L

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-42
SLIDE 42

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proof of L¨

  • bs Theorem

Let A be any sentence. L is Prov L ⊃ A. T ⊢ L ⊃ (Prov L ⊃ A) T ⊢ Prov L ⊃ (Prov L ⊃ A) T ⊢ Prov L ⊃ Prov Prov L ⊃ A T ⊢ Prov L ⊃ (Prov L ⊃ A) T ⊢ Prov L ⊃ A T ⊢ L T ⊢ Prov L T ⊢ A

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-43
SLIDE 43

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Modelling L¨

  • bs Principle

Let M be a finite collapsed model of the standard model

  • f arithmetic

Let T be Th(M)

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-44
SLIDE 44

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Modelling L¨

  • bs Principle

Let M be a finite collapsed model of the standard model

  • f arithmetic

Let T be Th(M) Everything true in the standard model is in T T is decidable

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-45
SLIDE 45

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Modelling L¨

  • bs Principle

Let M be a finite collapsed model of the standard model

  • f arithmetic

Let T be Th(M) Everything true in the standard model is in T T is decidable Let Prov be the arithmetic formula that defines T in the standard model

[3] if A ∈ T, Prov A is true in the standard model, and so is in T [4] if A / ∈ T, ¬Prov A is true in the standard model, and so is in T

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-46
SLIDE 46

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Take P as Prov

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-47
SLIDE 47

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Take P as Prov That ⊢ A implies ⊢ P A is immediate

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-48
SLIDE 48

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Take P as Prov That ⊢ A implies ⊢ P A is immediate . For ⊢ ¬P A ∨ A :

Either A ∈ T of A / ∈ T. In the first case, ¬Prov A ∨ A ∈ T. In the second case, ¬Prov A ∈ T So ¬Prov A ∨ A ∈ T.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-49
SLIDE 49

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Take P as Prov That ⊢ A implies ⊢ P A is immediate . For ⊢ ¬P A ∨ A :

Either A ∈ T of A / ∈ T. In the first case, ¬Prov A ∨ A ∈ T. In the second case, ¬Prov A ∈ T So ¬Prov A ∨ A ∈ T.

Moreover, unless the collapsed model is one in which everything is identified with 0, T is non-trivial.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-50
SLIDE 50

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Naive Arithmetic and Axiomatizability

Is Naive Arithmetic axiomatizable?

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-51
SLIDE 51

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Naive Arithmetic and Axiomatizability

Is Naive Arithmetic axiomatizable? Learning how to prove things in arithmetic is a skill that is taught and learned.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-52
SLIDE 52

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Naive Arithmetic and Axiomatizability

Is Naive Arithmetic axiomatizable? Learning how to prove things in arithmetic is a skill that is taught and learned. The assumption that the canons of naive proof are axiomatic is the most natural explanation of this fact.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-53
SLIDE 53

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Naive Arithmetic and Axiomatizability

Is Naive Arithmetic axiomatizable? Learning how to prove things in arithmetic is a skill that is taught and learned. The assumption that the canons of naive proof are axiomatic is the most natural explanation of this fact. Any other explanation would make the grasp of the canons something of a mystery for human cognition.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-54
SLIDE 54

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Practical Consistency

Let the least inconsistent number be n

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-55
SLIDE 55

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Practical Consistency

Let the least inconsistent number be n The fragment of arithmetic with quantifiers bounded to numbers less than n is consistent.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-56
SLIDE 56

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Practical Consistency

Let the least inconsistent number be n The fragment of arithmetic with quantifiers bounded to numbers less than n is consistent. n could be inordinately large

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-57
SLIDE 57

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proving Non-Triviality

Let T be any complete axiomatic arithmetic such that T ⊢ 0 = 1

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-58
SLIDE 58

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proving Non-Triviality

Let T be any complete axiomatic arithmetic such that T ⊢ 0 = 1 Then for every n, ¬B(n, 0 = 1) is true in the standard model

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-59
SLIDE 59

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proving Non-Triviality

Let T be any complete axiomatic arithmetic such that T ⊢ 0 = 1 Then for every n, ¬B(n, 0 = 1) is true in the standard model So ¬∃xB(x, 0 = 1) is true in the standard model

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-60
SLIDE 60

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proving Non-Triviality

Let T be any complete axiomatic arithmetic such that T ⊢ 0 = 1 Then for every n, ¬B(n, 0 = 1) is true in the standard model So ¬∃xB(x, 0 = 1) is true in the standard model That is, ¬Prov 0 = 1 is true in the standard model.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-61
SLIDE 61

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Proving Non-Triviality

Let T be any complete axiomatic arithmetic such that T ⊢ 0 = 1 Then for every n, ¬B(n, 0 = 1) is true in the standard model So ¬∃xB(x, 0 = 1) is true in the standard model That is, ¬Prov 0 = 1 is true in the standard model. So T ⊢ ¬Prov 0 = 1.

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness
slide-62
SLIDE 62

G¨

  • del’s

Theorem: Inconsistency vs Incomplete- ness Graham Priest Introduction: the Standard View G¨

  • del’s Proof

The Inconsistency

  • f Arithmetic

Non-Triviality Naive Arithmetic and Axiomati- zability Coda: G¨

  • del’s

Second Incom- pleteness Theorem

Fin

Graham Priest G¨

  • del’s Theorem: Inconsistency vs Incompleteness