[PPT] - Remarks on Gdels Incomplentess Theorems SATO Kentaro * PowerPoint Presentation

SLIDE 1

Remarks on Gödel’s Incomplentess Theorems

SATO Kentaro*

sato@inf.unibe.ch

SGSLPS Autumn 2016 ————————————————————————————- *His research is supported by John Templeton Foundation

– p. 1

SLIDE 2

Quiz 1

Gödel’s Completeness Theorem (1929):

The first order classical logic is complete.

– p. 2

SLIDE 3

Quiz 1

Gödel’s Completeness Theorem (1929):

The first order classical logic is complete. Henkin (1947) strengthened: Any theory over the first order classical logic is complete.

– p. 2

SLIDE 4

Quiz 1

Gödel’s Completeness Theorem (1929):

The first order classical logic is complete. Henkin (1947) strengthened: Any theory over the first order classical logic is complete. In particular: Peano Arithmetic PA is complete.

– p. 2

SLIDE 5

Quiz 1

Gödel’s Completeness Theorem (1929):

The first order classical logic is complete. Henkin (1947) strengthened: Any theory over the first order classical logic is complete. In particular: Peano Arithmetic PA is complete.

Gödel’s 1st Incompleteness Theorem (1931):

PA is incomplete.

– p. 2

SLIDE 6

Quiz 1 — Which is correct?

Gödel’s Completeness Theorem (1929):

The first order classical logic is complete. Henkin (1947) strengthened: Any theory over the first order classical logic is complete. In particular: Peano Arithmetic PA is complete.

Gödel’s 1st Incompleteness Theorem (1931):

PA is incomplete.

– p. 2

SLIDE 7

Quiz 2

Hilbert’s Programme looks for:

a complete and decidable axiomatization

f real numbers.

– p. 3

SLIDE 8

Quiz 2

Hilbert’s Programme looks for:

a complete and decidable axiomatization

f real numbers.

Gödel Incompleteness Theorem answers: “impossible”.

– p. 3

SLIDE 9

Quiz 2

Hilbert’s Programme looks for:

a complete and decidable axiomatization

f real numbers.

Gödel Incompleteness Theorem answers: “impossible”.

Tarski’s Theorem (1951):

quantifier elimination of real closed field.

– p. 3

SLIDE 10

Quiz 2

Hilbert’s Programme looks for:

a complete and decidable axiomatization

f real numbers.

Gödel Incompleteness Theorem answers: “impossible”.

Tarski’s Theorem (1951):

quantifier elimination of real closed field. As a consequence, it yields: a complete and decidable axiomatization

f (R, 0, 1, −, +, ·, <).

– p. 3

SLIDE 11

Quiz 2 — Which is correct?

Hilbert’s Programme looks for:

a complete and decidable axiomatization

f real numbers.

Gödel Incompleteness Theorem answers: “impossible”.

Tarski’s Theorem (1951):

quantifier elimination of real closed field. As a consequence, it yields: a complete and decidable axiomatization

f (R, 0, 1, −, +, ·, <).

– p. 3

SLIDE 12

Quiz 3

Gödel 2nd Incompleteness (1931):

PA cannot prove a sentence which represents the consistency of PA.

– p. 4

SLIDE 13

Quiz 3

Gödel 2nd Incompleteness (1931):

PA cannot prove a sentence which represents the consistency of PA.

Kreisel’s Remark (1960):

PA does prove a sentence which represents the consistency of PA.

– p. 4

SLIDE 14

Quiz 3 — Which is correct?

Gödel 2nd Incompleteness (1931):

PA cannot prove a sentence which represents the consistency of PA.

Kreisel’s Remark (1960):

PA does prove a sentence which represents the consistency of PA.

– p. 4

SLIDE 15

Outline

1. Know the statement correctly (40min):
the notions in the statement correctly;
the preconditions correctly;
counterexamples to preconditions?

– p. 5

SLIDE 16

Outline

1. Know the statement correctly (40min):
the notions in the statement correctly;
the preconditions correctly;
counterexamples to preconditions?
2. A brief look at the proofs (15min):
ω-consistency;
Gödel sentence vs. Rosser sentence;
Kreisel’s remark;
Loeb’s derivability conditions;

– p. 5

SLIDE 17

Outline

1. Know the statement correctly (40min):
the notions in the statement correctly;
the preconditions correctly;
counterexamples to preconditions?
2. A brief look at the proofs (15min):
ω-consistency;
Gödel sentence vs. Rosser sentence;
Kreisel’s remark;
Loeb’s derivability conditions;
3. Connection to the present-day researches (5min):
Gödel hierarchy;
my own contributions.

– p. 5

SLIDE 18

1. Know the Statement Correctly

– p. 6

SLIDE 19

The Statement

If a first order theory T satisfies the following:

...
...
...

then the following hold: 1st incompleteness: T is incomplete; 2nd incompleteness: T cannot prove a sentence which represents the consistency of T.

– p. 7

SLIDE 20

The Statement

If a first order theory T satisfies the following:

...
...
...

then the following hold: 1st incompleteness: T is incomplete;

– p. 7

SLIDE 21

The Statement

If a first order theory T satisfies the following:

...
...
...

then the following hold: 1st incompleteness: T is not complete.

– p. 7

SLIDE 22

Three Completenesses

Semantical Completeness “provable” ⇔ “true in any model”:

(Weak)

⊢ ϕ ⇐ ⇒ | = ϕ;

(Strong)

Γ ⊢ ϕ ⇐ ⇒ Γ | = ϕ. Negation Completeness Arithmetical Completeness

– p. 8

SLIDE 23

Three Completenesses

Semantical Completeness “provable” ⇔ “true in any model”:

(Weak)

⊢ ϕ ⇐ ⇒ | = ϕ;

(Strong)

Γ ⊢ ϕ ⇐ ⇒ Γ | = ϕ. Negation Completeness T can prove or disprove any sentence in LT:

T ⊢ ϕ or T ⊢ ¬ϕ for any sentence ϕ ∈ LT.

Arithmetical Completeness

– p. 8

SLIDE 24

Three Completenesses

Semantical Completeness “provable” ⇔ “true in any model”:

(Weak)

⊢ ϕ ⇐ ⇒ | = ϕ;

(Strong)

Γ ⊢ ϕ ⇐ ⇒ Γ | = ϕ. Negation Completeness T can prove or disprove any sentence in LT:

T ⊢ ϕ or T ⊢ ¬ϕ for any sentence ϕ ∈ LT.

Arithmetical Completeness “provable” ⇔ “true in the intended model”:

T ⊢ ϕ ⇐

⇒ N | = ϕ.

– p. 8

SLIDE 25

Three Completenesses

Semantical Completeness

Gödel-Henkin’s completeness theorem;
Kripke completeness

(modal logics, intuitionistic logic). Negation Completeness Arithmetical Completeness

– p. 9

SLIDE 26

Three Completenesses

Semantical Completeness

Gödel-Henkin’s completeness theorem;
Kripke completeness

(modal logics, intuitionistic logic). Negation Completeness

Gödel(-Rosser)’s 1st incompleteness theorem;
completeness of theories of

algebraic closed / real closed fields Arithmetical Completeness

– p. 9

SLIDE 27

Three Completenesses

Semantical Completeness

Gödel-Henkin’s completeness theorem;
Kripke completeness

(modal logics, intuitionistic logic). Negation Completeness

Gödel(-Rosser)’s 1st incompleteness theorem;
completeness of theories of

algebraic closed / real closed fields Arithmetical Completeness Σ0

1 completeness (of Q, PA, ZFC, etc.)

– p. 9

SLIDE 28

Quiz 1 — Which is correct?

Gödel’s Completeness Theorem (1929):

The first order classical logic is complete. Henkin (1947) strengthened: Any theory over the first order classical logic is complete. In particular: Peano Arithmetic PA is complete.

Gödel’s 1st Incompleteness Theorem (1931):

PA is incomplete.

– p. 10

SLIDE 29

The Statement (2)

If a first order theory T satisfies the following:

...
...
...

then the following hold: 1st incompleteness: T is not complete.

– p. 11

SLIDE 30

The Statement (2)

If a first order theory T satisfies the following:

T is consistent;
...
...

then the following hold: 1st incompleteness: T is not complete.

– p. 11

SLIDE 31

The Statement (2)

If a first order theory T satisfies the following:

T is consistent;
T is recursively axiomatizable;
...

then the following hold: 1st incompleteness: T is not complete.

– p. 11

SLIDE 32

The Statement (2)

If a first order theory T satisfies the following:

T is consistent;
T is recursively axiomatizable;
T essentially contains Robinson Arithmetic Q,

then the following hold: 1st incompleteness: T is not complete.

– p. 11

SLIDE 33

Consistency

A first order theory T is consistent iff

T ⊢ ⊥

and/or

T ⊢ ϕ for some ϕ

and/or

either T ⊢ ϕ or T ⊢ ¬ϕ for any ϕ, i.e.,

– p. 12

SLIDE 34

Consistency

A first order theory T is consistent iff

T ⊢ ⊥

and/or

T ⊢ ϕ for some ϕ

and/or

either T ⊢ ϕ or T ⊢ ¬ϕ for any ϕ, i.e.,

it’s not the case that T ⊢ ϕ and T ⊢ ¬ϕ.

– p. 12

SLIDE 35

Consistency

A first order theory T is consistent iff

T ⊢ ⊥

and/or

T ⊢ ϕ for some ϕ

and/or

either T ⊢ ϕ or T ⊢ ¬ϕ for any ϕ, i.e.,

it’s not the case that T ⊢ ϕ and T ⊢ ¬ϕ. If T is not consistent,

– p. 12

SLIDE 36

Consistency

A first order theory T is consistent iff

T ⊢ ⊥

and/or

T ⊢ ϕ for some ϕ

and/or

either T ⊢ ϕ or T ⊢ ¬ϕ for any ϕ, i.e.,

it’s not the case that T ⊢ ϕ and T ⊢ ¬ϕ. If T is not consistent,

T ⊢ ϕ for any ϕ;
hence either T ⊢ ϕ or T ⊢ ¬ϕ

(negation completeness).

– p. 12

SLIDE 37

The Statement (3)

If a first order theory T satisfies the following:

T is consistent;
T is recursively axiomatizable;
T essentially contains Robinson Arithmetic Q,

then the following hold: 1st incompleteness: T is not complete.

– p. 13

SLIDE 38

Recursive Axiomatizability

A first order theory T is recursively axiomatizable iff

there is Γ such that
Γ ⊢ ϕ ⇐

⇒ T ⊢ ϕ for any ϕ ∈ LT and

{

ϕ | ϕ ∈ Γ} is decidable;

– p. 14

SLIDE 39

Recursive Axiomatizability

A first order theory T is recursively axiomatizable iff

there is Γ such that
Γ ⊢ ϕ ⇐

⇒ T ⊢ ϕ for any ϕ ∈ LT and

{

ϕ | ϕ ∈ Γ} is decidable; and/or

there is Γ such that
Γ ⊢ ϕ ⇐

⇒ T ⊢ ϕ for any ϕ ∈ LT and

{

ϕ | ϕ ∈ Γ} is semi-decidable;

– p. 14

SLIDE 40

Recursive Axiomatizability

A first order theory T is recursively axiomatizable iff

there is Γ such that
Γ ⊢ ϕ ⇐

⇒ T ⊢ ϕ for any ϕ ∈ LT and

{

ϕ | ϕ ∈ Γ} is decidable; and/or

there is Γ such that
Γ ⊢ ϕ ⇐

⇒ T ⊢ ϕ for any ϕ ∈ LT and

{

ϕ | ϕ ∈ Γ} is semi-decidable; and/or

{

ϕ | T ⊢ ϕ} is semi-decidable.

– p. 14

SLIDE 41

Recursive Axiomatizability

A first order theory T is recursively axiomatizable iff

there is Γ such that
Γ ⊢ ϕ ⇐

⇒ T ⊢ ϕ for any ϕ ∈ LT and

{

ϕ | ϕ ∈ Γ} is decidable; and/or

there is Γ such that
Γ ⊢ ϕ ⇐

⇒ T ⊢ ϕ for any ϕ ∈ LT and

{

ϕ | ϕ ∈ Γ} is semi-decidable; and/or

{

ϕ | T ⊢ ϕ} is semi-decidable. Th(N) = {ϕ ∈ LPA | N | = ϕ} is negation complete.

– p. 14

SLIDE 42

Craig’s Theorem

If { ϕ | T ⊢ ϕ} is semi-decidable, then there is Γ such that

Γ ⊢ ϕ ⇐

⇒ T ⊢ ϕ for any ϕ ∈ LT and

{

ϕ | ϕ ∈ Γ} is decidable

– p. 15

SLIDE 43

Craig’s Theorem

If { ϕ | T ⊢ ϕ} is semi-decidable, then there is Γ such that

Γ ⊢ ϕ ⇐

⇒ T ⊢ ϕ for any ϕ ∈ LT and

{

ϕ | ϕ ∈ Γ} is decidable (Proof) Take a recursive predicate R such that T ⊢ ϕ ⇐ ⇒ ∃nR( ϕ , n) for any ϕ ∈ LT.

– p. 15

SLIDE 44

Craig’s Theorem

If { ϕ | T ⊢ ϕ} is semi-decidable, then there is Γ such that

Γ ⊢ ϕ ⇐

⇒ T ⊢ ϕ for any ϕ ∈ LT and

{

ϕ | ϕ ∈ Γ} is decidable (Proof) Take a recursive predicate R such that T ⊢ ϕ ⇐ ⇒ ∃nR( ϕ , n) for any ϕ ∈ LT. Define the following recursive set of axioms Γ = {ψ | (∃n, ϕ < ψ)(R( ϕ , n) & ψ ≡ ϕ∧...∧ϕ)}.

– p. 15

SLIDE 45

Craig’s Theorem

If { ϕ | T ⊢ ϕ} is semi-decidable, then there is Γ such that

Γ ⊢ ϕ ⇐

⇒ T ⊢ ϕ for any ϕ ∈ LT and

{

ϕ | ϕ ∈ Γ} is decidable (Proof) Take a recursive predicate R such that T ⊢ ϕ ⇐ ⇒ ∃nR( ϕ , n) for any ϕ ∈ LT. Define the following recursive set of axioms Γ = {ψ | (∃n, ϕ < ψ)(R( ϕ , n) & ψ ≡ ϕ∧...∧ϕ)}.

ψ ∈ Γ ⇒ T ⊢ ϕ & ψ ≡ ϕ∧...∧ϕ ⇒ T ⊢ ψ;
T ⊢ ϕ ⇒ ∃nR(

ϕ , n) ⇒ ϕ∧...∧ϕ

n+1

∈ Γ ⇒ Γ ⊢ ϕ.

– p. 15

SLIDE 46

Henkin Construction

Henkin’s Lemma: If Γ ⊢ ⊥ then there is maximal consistent ∆ ⊇ Γ.

– p. 16

SLIDE 47

Henkin Construction

Henkin’s Lemma: If Γ ⊢ ⊥ then there is maximal consistent ∆ ⊇ Γ. (Proof) Let ϕn’s enumerate all L formulae. Define Γn+1 := Γn if Γn ∪ {ϕn} ⊢ ⊥ Γn ∪ {ϕn} if Γn ∪ {ϕn} ⊢ ⊥. starting from Γ0 := Γ. Take ∆ :=

n∈ω Γn.

– p. 16

SLIDE 48

Henkin Construction

Henkin’s Lemma: If Γ ⊢ ⊥ then there is maximal consistent ∆ ⊇ Γ. (Proof) Let ϕn’s enumerate all L formulae. Define Γn+1 := Γn if Γn ∪ {ϕn} ⊢ ⊥ Γn ∪ {ϕn} if Γn ∪ {ϕn} ⊢ ⊥. starting from Γ0 := Γ. Take ∆ :=

n∈ω Γn.

Note:

The theory generated by ∆ is negation complete: either ϕ ∈ ∆ or ¬ϕ ∈ ∆ holds for any ϕ ∈ L.

– p. 16

SLIDE 49

The Statement (4)

If a first order theory T satisfies the following:

T is consistent;
T is recursively axiomatizable;
T essentially contains Robinson Arithmetic Q,

then the following hold: 1st incompleteness: T is not complete.

– p. 17

SLIDE 50

Robinson Arithmetic Q

Language (function) 0; S(-); +, ·; (relation) <. Axioms 1. ¬(S(x) = 0);

2. S(x) = S(y) → x = y;
3. x = 0 ∨ ∃y(x = S(y));
4. x+0 = x; and x+S(y) = S(x+y);
5. x·0 = 0; and x·S(y) = (x·y)+x.;
6. x < y ↔ ∃z(x+S(z) = y).

– p. 18

SLIDE 51

Robinson Arithmetic Q

Language (function) 0; S(-); +, ·; (relation) <. Axioms 1. ¬(S(x) = 0);

2. S(x) = S(y) → x = y;
3. x = 0 ∨ ∃y(x = S(y));
4. x+0 = x; and x+S(y) = S(x+y);
5. x·0 = 0; and x·S(y) = (x·y)+x.;
6. x < y ↔ ∃z(x+S(z) = y).

Remarks

first introduced by R. M. Robison in 1950 w/o <;
has no induction axiom (schema).

– p. 18

SLIDE 52

Theories containing Q

PA extends Q by induction scheme:

ϕ(0)∧∀x(ϕ(x) → ϕ(S(x)) → ∀xϕ(x) for ϕ ∈ LQ.

– p. 19

SLIDE 53

Theories containing Q

PA extends Q by induction scheme:

ϕ(0)∧∀x(ϕ(x) → ϕ(S(x)) → ∀xϕ(x) for ϕ ∈ LQ.

IΣn extends Q by induction for Σ0

n formulae:

1. Σ0

n = {∃xn∀xn−1...Qx1ϕ(

x) | ϕ ∈ ∆0

0} and

2. ϕ ∈ ∆0

0 iff all quantifiers in ϕ are bounded

(i.e., of the forms ∀x < t and ∃x < t).

– p. 19

SLIDE 54

Theories containing Q

PA extends Q by induction scheme:

ϕ(0)∧∀x(ϕ(x) → ϕ(S(x)) → ∀xϕ(x) for ϕ ∈ LQ.

IΣn extends Q by induction for Σ0

n formulae:

1. Σ0

n = {∃xn∀xn−1...Qx1ϕ(

x) | ϕ ∈ ∆0

0} and

2. ϕ ∈ ∆0

0 iff all quantifiers in ϕ are bounded

(i.e., of the forms ∀x < t and ∃x < t).

PRA extends Q by
1. LPRA := LQ ∪ {F | F ∈ PrimRec};
2. induction for quantifier-free LPRA formulae.

– p. 19

SLIDE 55

Theories containing Q

PA extends Q by induction scheme:

ϕ(0)∧∀x(ϕ(x) → ϕ(S(x)) → ∀xϕ(x) for ϕ ∈ LQ.

IΣn extends Q by induction for Σ0

n formulae:

1. Σ0

n = {∃xn∀xn−1...Qx1ϕ(

x) | ϕ ∈ ∆0

0} and

2. ϕ ∈ ∆0

0 iff all quantifiers in ϕ are bounded

(i.e., of the forms ∀x < t and ∃x < t).

PRA extends Q by
1. LPRA := LQ ∪ {F | F ∈ PrimRec};
2. induction for quantifier-free LPRA formulae.
ZFC extends Q ...

– p. 19

SLIDE 56

Theories containing Q

PA extends Q by induction scheme:

ϕ(0)∧∀x(ϕ(x) → ϕ(S(x)) → ∀xϕ(x) for ϕ ∈ LQ.

IΣn extends Q by induction for Σ0

n formulae:

1. Σ0

n = {∃xn∀xn−1...Qx1ϕ(

x) | ϕ ∈ ∆0

0} and

2. ϕ ∈ ∆0

0 iff all quantifiers in ϕ are bounded

(i.e., of the forms ∀x < t and ∃x < t).

PRA extends Q by
1. LPRA := LQ ∪ {F | F ∈ PrimRec};
2. induction for quantifier-free LPRA formulae.
ZFC extends Q ...

really? in which sense?

– p. 19

SLIDE 57

Interpretation

An interpretation I of L in L′ consists of:

an L′ formula υI(x), called universe;
for function f(

x) of L, an L′ formula f I(y, x);

for relation R(

x) of L, an L′ formula RI( x).

– p. 20

SLIDE 58

Interpretation

An interpretation I of L in L′ consists of:

an L′ formula υI(x), called universe;
for function f(

x) of L, an L′ formula f I(y, x);

for relation R(

x) of L, an L′ formula RI( x). Extend I to all L-terms and L-formulae:

if t(

x) ≡ f(t1( x), ..., tk( x)), then tI(y, x) ≡ ∃z1, ..., zk(

i≤k tiI(zi,

x) ∧ f I(y, z1, .., zn));

– p. 20

SLIDE 59

Interpretation

An interpretation I of L in L′ consists of:

an L′ formula υI(x), called universe;
for function f(

x) of L, an L′ formula f I(y, x);

for relation R(

x) of L, an L′ formula RI( x). Extend I to all L-terms and L-formulae:

if t(

x) ≡ f(t1( x), ..., tk( x)), then tI(y, x) ≡ ∃z1, ..., zk(

i≤k tiI(zi,

x) ∧ f I(y, z1, .., zn));

if ϕ ≡ R(t1(

x), ..., tk( x)), then ϕI ≡ ∃z1, ..., zk(

i≤k tiI(zi,

x) ∧ RI(z1, .., zn));

– p. 20

SLIDE 60

Interpretation

An interpretation I of L in L′ consists of:

an L′ formula υI(x), called universe;
for function f(

x) of L, an L′ formula f I(y, x);

for relation R(

x) of L, an L′ formula RI( x). Extend I to all L-terms and L-formulae:

if t(

x) ≡ f(t1( x), ..., tk( x)), then tI(y, x) ≡ ∃z1, ..., zk(

i≤k tiI(zi,

x) ∧ f I(y, z1, .., zn));

if ϕ ≡ R(t1(

x), ..., tk( x)), then ϕI ≡ ∃z1, ..., zk(

i≤k tiI(zi,

x) ∧ RI(z1, .., zn));

(ϕ∧ψ)I ≡ ϕI ∧ ψI; and (¬ϕ)I ≡ ¬ϕI;

– p. 20

SLIDE 61

Interpretation

An interpretation I of L in L′ consists of:

an L′ formula υI(x), called universe;
for function f(

x) of L, an L′ formula f I(y, x);

for relation R(

x) of L, an L′ formula RI( x). Extend I to all L-terms and L-formulae:

if t(

x) ≡ f(t1( x), ..., tk( x)), then tI(y, x) ≡ ∃z1, ..., zk(

i≤k tiI(zi,

x) ∧ f I(y, z1, .., zn));

if ϕ ≡ R(t1(

x), ..., tk( x)), then ϕI ≡ ∃z1, ..., zk(

i≤k tiI(zi,

x) ∧ RI(z1, .., zn));

(ϕ∧ψ)I ≡ ϕI ∧ ψI; and (¬ϕ)I ≡ ¬ϕI;
(∀yϕ(y))I ≡ ∀y(υI(y) → ϕ(y)I).

– p. 20

SLIDE 62

Interpretation 2

Given an interpretation I of L in L′.

I is an interpretation in an L′ theory T ′ iff
1. T ′ ⊢ ∃xυI(x);
2. T ′ ⊢ ∀

x(υI( x) → ∃!y(υI(y) ∧ f I(y, x))).

– p. 21

SLIDE 63

Interpretation 2

Given an interpretation I of L in L′.

I is an interpretation in an L′ theory T ′ iff
1. T ′ ⊢ ∃xυI(x);
2. T ′ ⊢ ∀

x(υI( x) → ∃!y(υI(y) ∧ f I(y, x))).

I is an interpretation of an L theory T in T ′ iff
1. (as above);
2. (as above);
3. if T ⊢ ϕ then T ′ ⊢ ϕI for any ϕ ∈ L.

– p. 21

SLIDE 64

Theories ess. containing Q

“T ′ ess. contains T” = “∃ interpretation of T in T ′”.

– p. 22

SLIDE 65

Theories ess. containing Q

“T ′ ess. contains T” = “∃ interpretation of T in T ′”.

ZFC essentially contains Q

by von Neumann interpretation v:

1. υv(x) ≡ “x is a finite von Neumann ordinal”;
2. Sv(y, x) ≡ y = x ∪ {x},

etc.;

– p. 22

SLIDE 66

Theories ess. containing Q

“T ′ ess. contains T” = “∃ interpretation of T in T ′”.

ZFC essentially contains Q

by von Neumann interpretation v:

1. υv(x) ≡ “x is a finite von Neumann ordinal”;
2. Sv(y, x) ≡ y = x ∪ {x},

etc.;

modal extensions of PA (directly) contains Q;

– p. 22

SLIDE 67

Theories ess. containing Q

“T ′ ess. contains T” = “∃ interpretation of T in T ′”.

ZFC essentially contains Q

by von Neumann interpretation v:

1. υv(x) ≡ “x is a finite von Neumann ordinal”;
2. Sv(y, x) ≡ y = x ∪ {x},

etc.;

modal extensions of PA (directly) contains Q;
Heyting Arithmetic HA (ess.) contains Q by
HA literally extends PA in ∧, ¬, ∀,

with extra-operators ∨, ∃ (like modality);

– p. 22

SLIDE 68

Theories ess. containing Q

“T ′ ess. contains T” = “∃ interpretation of T in T ′”.

ZFC essentially contains Q

by von Neumann interpretation v:

1. υv(x) ≡ “x is a finite von Neumann ordinal”;
2. Sv(y, x) ≡ y = x ∪ {x},

etc.;

modal extensions of PA (directly) contains Q;
Heyting Arithmetic HA (ess.) contains Q by
HA literally extends PA in ∧, ¬, ∀,

with extra-operators ∨, ∃ (like modality);

relaxing the notion of interpretation so that

double negation translation N is included: (ϕ∨ψ)N ≡ ¬(¬ϕN∧¬ψN); (∃xϕ(x))N ≡ ¬∀x¬ϕ(x)N; etc.

– p. 22

SLIDE 69

Presburger Arithmetic PresA

Language LPresA = {0, S, +}; Axioms 1. ¬(S(x) = 0);

2. S(x) = S(y) → x = y;
3. x = 0 ∨ ∃y(x = S(y));
4. x+0 = x; and x+S(y) = S(x+y);
5. induction for all LPresA formulae.

– p. 23

SLIDE 70

Presburger Arithmetic PresA

Language LPresA = {0, S, +}; Axioms 1. ¬(S(x) = 0);

2. S(x) = S(y) → x = y;
3. x = 0 ∨ ∃y(x = S(y));
4. x+0 = x; and x+S(y) = S(x+y);
5. induction for all LPresA formulae.

Remarks

essentially, the ·-free fragment of PA;

– p. 23

SLIDE 71

Presburger Arithmetic PresA

Language LPresA = {0, S, +}; Axioms 1. ¬(S(x) = 0);

2. S(x) = S(y) → x = y;
3. x = 0 ∨ ∃y(x = S(y));
4. x+0 = x; and x+S(y) = S(x+y);
5. induction for all LPresA formulae.

Remarks

essentially, the ·-free fragment of PA;
introduced by M. Presburger in 1929;

– p. 23

SLIDE 72

Presburger Arithmetic PresA

Language LPresA = {0, S, +}; Axioms 1. ¬(S(x) = 0);

2. S(x) = S(y) → x = y;
3. x = 0 ∨ ∃y(x = S(y));
4. x+0 = x; and x+S(y) = S(x+y);
5. induction for all LPresA formulae.

Remarks

essentially, the ·-free fragment of PA;
introduced by M. Presburger in 1929;
proven by him to be complete, i.e.,

PreA ⊢ ϕ ⇐ ⇒ N | = ϕ for any ϕ ∈ LPresA;

– p. 23

SLIDE 73

Presburger Arithmetic PresA

Language LPresA = {0, S, +}; Axioms 1. ¬(S(x) = 0);

2. S(x) = S(y) → x = y;
3. x = 0 ∨ ∃y(x = S(y));
4. x+0 = x; and x+S(y) = S(x+y);
5. induction for all LPresA formulae.

Remarks

essentially, the ·-free fragment of PA;
introduced by M. Presburger in 1929;
proven by him to be complete, i.e.,

PreA ⊢ ϕ ⇐ ⇒ N | = ϕ for any ϕ ∈ LPresA;

hence not essentially contains Q.

– p. 23

SLIDE 74

Theory of real closed fields RCF

Language LRCF := {0, 1, −, +, ·, <}; Axioms 1. x+0 = x; x+(−x) = 0; x+y = y+x;

2. x·0 = 0;

x·(y+z) = x·y+x·z; x·y = y·x;

3. x < y → x+z < y+z;

x > 0 ∧ y > 0 → x·y > 0;

4. x > 0 → ∃y(x = y·y);
5. ∀x2n+1...x0(x2n+1 = 0 → ∃y(

i≤2n+1 xi·yi = 0)).

– p. 24

SLIDE 75

Theory of real closed fields RCF

Language LRCF := {0, 1, −, +, ·, <}; Axioms 1. x+0 = x; x+(−x) = 0; x+y = y+x;

2. x·0 = 0;

x·(y+z) = x·y+x·z; x·y = y·x;

3. x < y → x+z < y+z;

x > 0 ∧ y > 0 → x·y > 0;

4. x > 0 → ∃y(x = y·y);
5. ∀x2n+1...x0(x2n+1 = 0 → ∃y(

i≤2n+1 xi·yi = 0)).

Remarks

proven by Tarski (1951) to admit

quantifier-elimination; and so

RFC ⊢ ϕ ⇐

⇒ (R, 0, 1, −, +, ·, <) | = ϕ;

– p. 24

SLIDE 76

Theory of real closed fields RCF

Language LRCF := {0, 1, −, +, ·, <}; Axioms 1. x+0 = x; x+(−x) = 0; x+y = y+x;

2. x·0 = 0;

x·(y+z) = x·y+x·z; x·y = y·x;

3. x < y → x+z < y+z;

x > 0 ∧ y > 0 → x·y > 0;

4. x > 0 → ∃y(x = y·y);
5. ∀x2n+1...x0(x2n+1 = 0 → ∃y(

i≤2n+1 xi·yi = 0)).

Remarks

proven by Tarski (1951) to admit

quantifier-elimination; and so

RFC ⊢ ϕ ⇐

⇒ (R, 0, 1, −, +, ·, <) | = ϕ;

hence not essentially contains Q.

– p. 24

SLIDE 77

Quiz 2 — Which is correct?

Hilbert’s Programme looks for:

a complete and decidable axiomatization

f real numbers.

Gödel Incompleteness Theorem answers: “impossible”.

Tarski’s Theorem (1951):

quantifier elimination of real closed field. As a consequence, it yields: a complete and decidable axiomatization

f (R, 0, 1, −, +, ·, <).

– p. 25

SLIDE 78

The Statement (5)

If a first order theory T satisfies the following:

T is consistent;
T is recursively axiomatizable;
T essentially contains Robinson Arithmetic Q,

then the following hold: 1st incompleteness: T is not complete;

– p. 26

SLIDE 79

The Statement (5)

If a first order theory T satisfies the following:

T is consistent;
T is recursively axiomatizable;
T essentially contains Robinson Arithmetic Q,

then the following hold: 1st incompleteness: T is not complete; 2nd incompleteness: T cannot prove a sentence which represents the consistency of T.

– p. 26

SLIDE 80

Nelson’s trick

There is an interpretation of IΣ0 + Ω1 in Q.

Main idea: define an LQ formula W(x) which

intuitively means “< is well-founded below x”;

– p. 27

SLIDE 81

Nelson’s trick

There is an interpretation of IΣ0 + Ω1 in Q.

Main idea: define an LQ formula W(x) which

intuitively means “< is well-founded below x”;

Nelson’s interpretation n should be
1. υn(x) ≡ W(x);
2. Sn(y, x) ≡ y = S(x);

+n(z, x, y) ≡ z = x+y; ·n(z, x, y) ≡ z = x·y.

3. =n(x, y) ≡ x = y;

<n(x, y) ≡ x < y.

– p. 27

SLIDE 82

Nelson’s trick

There is an interpretation of IΣ0 + Ω1 in Q.

Main idea: define an LQ formula W(x) which

intuitively means “< is well-founded below x”;

Nelson’s interpretation n should be
1. υn(x) ≡ W(x);
2. Sn(y, x) ≡ y = S(x);

+n(z, x, y) ≡ z = x+y; ·n(z, x, y) ≡ z = x·y.

3. =n(x, y) ≡ x = y;

<n(x, y) ≡ x < y.

Compare to ϕ → ϕWF in set theory.

– p. 27

SLIDE 83

Nelson’s trick

There is an interpretation of IΣ0 + Ω1 in Q.

Main idea: define an LQ formula W(x) which

intuitively means “< is well-founded below x”;

Nelson’s interpretation n should be
1. υn(x) ≡ W(x);
2. Sn(y, x) ≡ y = S(x);

+n(z, x, y) ≡ z = x+y; ·n(z, x, y) ≡ z = x·y.

3. =n(x, y) ≡ x = y;

<n(x, y) ≡ x < y.

Compare to ϕ → ϕWF in set theory.

As a consequence, “T ess. contains Q” ⇐ ⇒ “T ess. contains IΣ0+Ω1”

– p. 27

SLIDE 84

Numeralwise representation

R ⊆ ωn is numeralwise represented by ϕ( x) iff

Q ⊢ ϕ(k1, ..., kn) ⇐

⇒ R(k1, ..., kn) and

Q ⊢ ¬ϕ(k1, ..., kn) ⇐

⇒ ¬R(k1, ..., kn), where k := S(...(S

k

(0)...).

– p. 28

SLIDE 85

Numeralwise representation

R ⊆ ωn is numeralwise represented by ϕ( x) iff

Q ⊢ ϕ(k1, ..., kn) ⇐

⇒ R(k1, ..., kn) and

Q ⊢ ¬ϕ(k1, ..., kn) ⇐

⇒ ¬R(k1, ..., kn), where k := S(...(S

k

(0)...). We have the following LQ formulae (Σ0

1 completeness):

Q ⊢ neg(

ϕ , k) ⇐ ⇒ k = ¬ϕ and Q ⊢ ¬neg( ϕ , k) ⇐ ⇒ k = ¬ϕ ;

– p. 28

SLIDE 86

Numeralwise representation

R ⊆ ωn is numeralwise represented by ϕ( x) iff

Q ⊢ ϕ(k1, ..., kn) ⇐

⇒ R(k1, ..., kn) and

Q ⊢ ¬ϕ(k1, ..., kn) ⇐

⇒ ¬R(k1, ..., kn), where k := S(...(S

k

(0)...). We have the following LQ formulae (Σ0

1 completeness):

Q ⊢ neg(

ϕ , k) ⇐ ⇒ k = ¬ϕ and Q ⊢ ¬neg( ϕ , k) ⇐ ⇒ k = ¬ϕ ;

Q ⊢ PrfT(

Λ , ϕ ) ⇐ ⇒ Λ is a T-proof of ϕ Q ⊢ ¬PrfT( Λ , ϕ ) ⇐ ⇒ Λ is not a T-proof of ϕ (if T is recursively axiomatizable).

– p. 28

SLIDE 87

Numeralwise representation

R ⊆ ωn is numeralwise represented by ϕ( x) iff

Q ⊢ ϕ(k1, ..., kn) ⇐

⇒ R(k1, ..., kn) and

Q ⊢ ¬ϕ(k1, ..., kn) ⇐

⇒ ¬R(k1, ..., kn), where k := S(...(S

k

(0)...). We have the following LQ formulae (Σ0

1 completeness):

Q ⊢ neg(

ϕ , k) ⇐ ⇒ k = ¬ϕ and Q ⊢ ¬neg( ϕ , k) ⇐ ⇒ k = ¬ϕ ;

Q ⊢ PrfT(

Λ , ϕ ) ⇐ ⇒ Λ is a T-proof of ϕ Q ⊢ ¬PrfT( Λ , ϕ ) ⇐ ⇒ Λ is not a T-proof of ϕ (if T is recursively axiomatizable). Then it is natural to define Con(T) :≡ ¬∃xPrfT(x, ⊥ ).

– p. 28

SLIDE 88

Ambiguity

Even if the following hold for all Λ and ϕ:

Q ⊢ PrfT(

Λ , ϕ ) ⇐ ⇒ Λ is a T-proof of ϕ and Q ⊢ ¬PrfT( Λ , ϕ ) ⇐ ⇒ Λ is not a T-proof of ϕ;

Q ⊢ Prf∗

T(

Λ , ϕ ) ⇐ ⇒ Λ is a T-proof of ϕ and Q ⊢ ¬Prf∗

T(

Λ , ϕ ) ⇐ ⇒ Λ is not a T-proof of ϕ;

– p. 29

SLIDE 89

Ambiguity

Even if the following hold for all Λ and ϕ:

Q ⊢ PrfT(

Λ , ϕ ) ⇐ ⇒ Λ is a T-proof of ϕ and Q ⊢ ¬PrfT( Λ , ϕ ) ⇐ ⇒ Λ is not a T-proof of ϕ;

Q ⊢ Prf∗

T(

Λ , ϕ ) ⇐ ⇒ Λ is a T-proof of ϕ and Q ⊢ ¬Prf∗

T(

Λ , ϕ ) ⇐ ⇒ Λ is not a T-proof of ϕ; we do not have

Q ⊢ ∀x, y(PrfT(x, y) ↔ Prf∗

T(x, y)), nor

– p. 29

SLIDE 90

Ambiguity

Even if the following hold for all Λ and ϕ:

Q ⊢ PrfT(

Λ , ϕ ) ⇐ ⇒ Λ is a T-proof of ϕ and Q ⊢ ¬PrfT( Λ , ϕ ) ⇐ ⇒ Λ is not a T-proof of ϕ;

Q ⊢ Prf∗

T(

Λ , ϕ ) ⇐ ⇒ Λ is a T-proof of ϕ and Q ⊢ ¬Prf∗

T(

Λ , ϕ ) ⇐ ⇒ Λ is not a T-proof of ϕ; we do not have

Q ⊢ ∀x, y(PrfT(x, y) ↔ Prf∗

T(x, y)), nor

Q ⊢ Con(T) ↔ Con∗(T),

where Con∗(T) :≡ ¬∃xPrf∗

T(x,

⊥ ).

– p. 29

SLIDE 91

Ambiguity

Even if the following hold for all Λ and ϕ:

Q ⊢ PrfT(

Λ , ϕ ) ⇐ ⇒ Λ is a T-proof of ϕ and Q ⊢ ¬PrfT( Λ , ϕ ) ⇐ ⇒ Λ is not a T-proof of ϕ;

Q ⊢ Prf∗

T(

Λ , ϕ ) ⇐ ⇒ Λ is a T-proof of ϕ and Q ⊢ ¬Prf∗

T(

Λ , ϕ ) ⇐ ⇒ Λ is not a T-proof of ϕ; we do not have

Q ⊢ ∀x, y(PrfT(x, y) ↔ Prf∗

T(x, y)), nor

Q ⊢ Con(T) ↔ Con∗(T),

where Con∗(T) :≡ ¬∃xPrf∗

T(x,

⊥ ). The point here: T ⊢ ϕ(k) for all k ∈ ω ⇒ T ⊢ ∀xϕ(x).

– p. 29

SLIDE 92

Quiz 3 — Which is correct?

Gödel 2nd Incompleteness (1931):

PA cannot prove a sentence which represents the consistency of PA.

Kreisel’s Remark (1960):

PA does prove a sentence which represents the consistency of PA.

– p. 30

SLIDE 93

2. A Brief Look at the Proofs

– p. 31

SLIDE 94

Rosser’s trick

Given PrfT such that, for all Λ and ϕ,

Q ⊢ PrfT(

Λ , ϕ ) ⇐ ⇒ Λ is a T-proof of ϕ,

Q ⊢ ¬PrfT(

Λ , ϕ ) ⇐ ⇒ Λ is not a T-proof of ϕ,

– p. 32

SLIDE 95

Rosser’s trick

Given PrfT such that, for all Λ and ϕ,

Q ⊢ PrfT(

Λ , ϕ ) ⇐ ⇒ Λ is a T-proof of ϕ,

Q ⊢ ¬PrfT(

Λ , ϕ ) ⇐ ⇒ Λ is not a T-proof of ϕ, we can define Prf∗

T by

Prf∗

T(x, u) : ≡ Prf(x, u)∧

(∀z < x)∀v¬(neg(u, v) ∧ Prf(z, v)).

– p. 32

SLIDE 96

Rosser’s trick

Given PrfT such that, for all Λ and ϕ,

Q ⊢ PrfT(

Λ , ϕ ) ⇐ ⇒ Λ is a T-proof of ϕ,

Q ⊢ ¬PrfT(

Λ , ϕ ) ⇐ ⇒ Λ is not a T-proof of ϕ, we can define Prf∗

T by

Prf∗

T(x, u) : ≡ Prf(x, u)∧

(∀z < x)∀v¬(neg(u, v) ∧ Prf(z, v)). Then Q ⊢ Prf∗

T(

Λ , ϕ ) ⇐ ⇒ Λ is a T-proof of ϕ ∧ there is no T-proof ∆ of ¬ϕ with ∆ < Λ

=

⇒ Λ is a T-proof of ϕ.

– p. 32

SLIDE 97

Rosser’s trick

Given PrfT such that, for all Λ and ϕ,

Q ⊢ PrfT(

Λ , ϕ ) ⇐ ⇒ Λ is a T-proof of ϕ,

Q ⊢ ¬PrfT(

Λ , ϕ ) ⇐ ⇒ Λ is not a T-proof of ϕ, we can define Prf∗

T by

Prf∗

T(x, u) : ≡ Prf(x, u)∧

(∀z < x)∀v¬(neg(u, v) ∧ Prf(z, v)). Then Q ⊢ Prf∗

T(

Λ , ϕ ) ⇐ ⇒ Λ is a T-proof of ϕ ∧ there is no T-proof ∆ of ¬ϕ with ∆ < Λ

=

⇒ Λ is a T-proof of ϕ. If T is consistent, ⇐ = also holds.

– p. 32

SLIDE 98

Rosser’s trick

Given PrfT such that, for all Λ and ϕ,

Q ⊢ PrfT(

Λ , ϕ ) ⇐ ⇒ Λ is a T-proof of ϕ,

Q ⊢ ¬PrfT(

Λ , ϕ ) ⇐ ⇒ Λ is not a T-proof of ϕ, we can define Prf∗

T by

Prf∗

T(x, u) : ≡ Prf(x, u)∧

(∀z < x)∀v¬(neg(u, v) ∧ Prf(z, v)). Then Q ⊢ ¬Prf∗

T(

Λ , ϕ ) ⇐ ⇒ Λ is not a T-proof of ϕ ∨ there is a T-proof ∆ of ¬ϕ with ∆ < Λ

⇐

= Λ is not a T-proof of ϕ.

– p. 32

SLIDE 99

Rosser’s trick

Given PrfT such that, for all Λ and ϕ,

Q ⊢ PrfT(

Λ , ϕ ) ⇐ ⇒ Λ is a T-proof of ϕ,

Q ⊢ ¬PrfT(

Λ , ϕ ) ⇐ ⇒ Λ is not a T-proof of ϕ, we can define Prf∗

T by

Prf∗

T(x, u) : ≡ Prf(x, u)∧

(∀z < x)∀v¬(neg(u, v) ∧ Prf(z, v)). Then Q ⊢ ¬Prf∗

T(

Λ , ϕ ) ⇐ ⇒ Λ is not a T-proof of ϕ ∨ there is a T-proof ∆ of ¬ϕ with ∆ < Λ

⇐

= Λ is not a T-proof of ϕ. If T is consistent, = ⇒ also holds.

– p. 32

SLIDE 100

Kreisel’s remark (1960)

Since there is a proof ∆ of ¬⊥, if T is consistent, Q ⊢ (∀x < ∆)¬Prf(x, ⊥ ).

– p. 33

SLIDE 101

Kreisel’s remark (1960)

Since there is a proof ∆ of ¬⊥, if T is consistent, Q ⊢ (∀x < ∆)¬Prf(x, ⊥ ). Hence, Q proves Prf∗

T(x,

⊥ ) ≡ PrfT(x, ⊥ ) ∧ (∀z < x)∀v¬(neg( ⊥ , v) ∧ PrfT(z, v)) → ∀v¬(neg( ⊥ , v) ∧ PrfT(∆, v)) ↔ ¬PrfT(∆, ¬⊥) → ⊥.

– p. 33

SLIDE 102

Kreisel’s remark (1960)

Since there is a proof ∆ of ¬⊥, if T is consistent, Q ⊢ (∀x < ∆)¬Prf(x, ⊥ ). Hence, Q proves Prf∗

T(x,

⊥ ) ≡ PrfT(x, ⊥ ) ∧ (∀z < x)∀v¬(neg( ⊥ , v) ∧ PrfT(z, v)) → ∀v¬(neg( ⊥ , v) ∧ PrfT(∆, v)) ↔ ¬PrfT(∆, ¬⊥) → ⊥. For any consistent recursively axiomatizable T, Q ⊢ Con∗(T).

– p. 33

SLIDE 103

The Statement (6)

If a first order theory T satisfies the following:

T is consistent;
T is recursively axiomatizable;
T essentially contains Robinson Arithmetic Q,

then the following hold: 1st incompleteness: T is not complete; 2nd incompleteness: T cannot prove a sentence which represents the consistency of T.

– p. 34

SLIDE 104

Gödel’s result

If a first order theory T satisfies the following:

T is ω-consistent;
T is recursively axiomatizable;
T essentially contains Robinson Arithmetic Q,

then the following hold: 1st incompleteness: T is not complete; 2nd incompleteness: T cannot prove a sentence which represents the consistency of T.

– p. 35

SLIDE 105

Gödel’s result

If a first order theory T satisfies the following:

T is ω-consistent;
T is recursively axiomatizable;
T essentially contains Robinson Arithmetic Q,

then the following hold: 1st incompleteness: T is not complete; 2nd incompleteness: T cannot prove a sentence which represents the consistency of T. T is called ω-consistent iff there is no ϕ(x) ∈ LT s.t.

T ⊢ ¬ϕ(k) for all k ∈ ω;
T ⊢ ∃xϕ(x).

– p. 35

SLIDE 106

Gödel’s Self-reference Lemma

Lemma For any ϕ(x) ∈ LQ, there is a LQ sentence θ s.t. Q ⊢ θ ↔ ϕ( θ).

– p. 36

SLIDE 107

Gödel’s Self-reference Lemma

Lemma For any ϕ(x) ∈ LQ, there is a LQ sentence θ s.t. Q ⊢ θ ↔ ϕ( θ). (proof) Take Subst(u, y, v) such that

Q ⊢ Subst(

τ(x) , t , k) ⇐ ⇒ k = τ(t) ;

Q ⊢ ¬Subst(

τ(x) , t , k) ⇐ ⇒ k = τ(t) ;

– p. 36

SLIDE 108

Gödel’s Self-reference Lemma

Lemma For any ϕ(x) ∈ LQ, there is a LQ sentence θ s.t. Q ⊢ θ ↔ ϕ( θ). (proof) Take Subst(u, y, v) such that

Q ⊢ Subst(

τ(x) , t , k) ⇐ ⇒ k = τ(t) ;

Q ⊢ ¬Subst(

τ(x) , t , k) ⇐ ⇒ k = τ(t) ; Let ρ(x) :≡ ∃u(Subst(x, x , u) ∧ ϕ(u)).

– p. 36

SLIDE 109

Gödel’s Self-reference Lemma

Lemma For any ϕ(x) ∈ LQ, there is a LQ sentence θ s.t. Q ⊢ θ ↔ ϕ( θ). (proof) Take Subst(u, y, v) such that

Q ⊢ Subst(

τ(x) , t , k) ⇐ ⇒ k = τ(t) ;

Q ⊢ ¬Subst(

τ(x) , t , k) ⇐ ⇒ k = τ(t) ; Let ρ(x) :≡ ∃u(Subst(x, x , u) ∧ ϕ(u)). For any τ(x), Q ⊢ ρ( τ(x) ) ↔ ϕ( τ( τ(x) ) ).

– p. 36

SLIDE 110

Gödel’s Self-reference Lemma

Lemma For any ϕ(x) ∈ LQ, there is a LQ sentence θ s.t. Q ⊢ θ ↔ ϕ( θ). (proof) Take Subst(u, y, v) such that

Q ⊢ Subst(

τ(x) , t , k) ⇐ ⇒ k = τ(t) ;

Q ⊢ ¬Subst(

τ(x) , t , k) ⇐ ⇒ k = τ(t) ; Let ρ(x) :≡ ∃u(Subst(x, x , u) ∧ ϕ(u)). For any τ(x), Q ⊢ ρ( τ(x) ) ↔ ϕ( τ( τ(x) ) ). Letting τ(x) ≡ ρ(x) and θ ≡ ρ( ρ(x) ), we have Q ⊢ ρ( ρ(x) ) ↔ ϕ( ρ( ρ(x) ) ).

– p. 36

SLIDE 111

Gödel’s 1st incompleteness

Theorem If T is ω-consistent, ...(omitted)..., then there is σ ∈ LQ s.t. T ⊢ σ and T ⊢ ¬σ.

– p. 37

SLIDE 112

Gödel’s 1st incompleteness

Theorem If T is ω-consistent, ...(omitted)..., then there is σ ∈ LQ s.t. T ⊢ σ and T ⊢ ¬σ. (Proof) By self-reference lemma, we have σ s.t. Q ⊢ σ ↔ ¬∃xPrfT(x, σ).

– p. 37

SLIDE 113

Gödel’s 1st incompleteness

Theorem If T is ω-consistent, ...(omitted)..., then there is σ ∈ LQ s.t. T ⊢ σ and T ⊢ ¬σ. (Proof) By self-reference lemma, we have σ s.t. Q ⊢ σ ↔ ¬∃xPrfT(x, σ). Suppose T ⊢ σ. There is a T-proof Λ of σ. Then Q ⊢ PrfT( Λ , σ), and Q ⊢ ∃xPrfT(x, σ). Thus Q ⊢ ¬σ, contradicting the consistency of T.

– p. 37

SLIDE 114

Gödel’s 1st incompleteness

Theorem If T is ω-consistent, ...(omitted)..., then there is σ ∈ LQ s.t. T ⊢ σ and T ⊢ ¬σ. (Proof) By self-reference lemma, we have σ s.t. Q ⊢ σ ↔ ¬∃xPrfT(x, σ). Suppose T ⊢ σ. There is a T-proof Λ of σ. Then Q ⊢ PrfT( Λ , σ), and Q ⊢ ∃xPrfT(x, σ). Thus Q ⊢ ¬σ, contradicting the consistency of T. Suppose T ⊢ ¬σ. Then T ⊢ σ by consistency. Thus Q ⊢ ¬PrfT(k, σ) for all k ∈ ω. However, T ⊢ ∃xPrfT(x, σ), and so T is ω-inconsistent.

– p. 37

SLIDE 115

Rosser’s enhancement

Theorem If T is consistent, ...(omitted)..., then there is σ ∈ LQ s.t. T ⊢ σ and T ⊢ ¬σ. (Proof) By self-reference lemma, we have σ s.t. Q ⊢ σ ↔ ¬∃xPrf∗

T(x,

σ). Suppose T ⊢ σ. There is a T-proof Λ of σ. Then Q ⊢ Prf∗

T(

Λ , σ), and Q ⊢ ∃xPrf∗

T(x,

σ). Thus Q ⊢ ¬σ, contradicting the consistency of T.

– p. 38

SLIDE 116

Rosser’s enhancement

Theorem If T is consistent, ...(omitted)..., then there is σ ∈ LQ s.t. T ⊢ σ and T ⊢ ¬σ. (Proof) By self-reference lemma, we have σ s.t. Q ⊢ σ ↔ ¬∃xPrf∗

T(x,

σ). Suppose T ⊢ σ. There is a T-proof Λ of σ. Then Q ⊢ Prf∗

T(

Λ , σ), and Q ⊢ ∃xPrf∗

T(x,

σ). Thus Q ⊢ ¬σ, contradicting the consistency of T. Suppose T ⊢ ¬σ. So Q ⊢ Prf∗

T(

∆ , ¬σ) for some ∆. Since T is consistent, Q ⊢ (∀x < ∆ )¬PrfT(x, σ). But T ⊢ ∃xPrf∗

T(x,

σ), i.e., T ⊢ ∃x(PrfT(x, σ) ∧ (∀y < x)¬PrfT(y, ¬σ)).

– p. 38

SLIDE 117

A dilemma

To obtain the incompleteness

without ω-consistency but only consistency, the key is Rosser’s modification Prf∗

T

for representing the notion “... is a proof of ...”;

– p. 39

SLIDE 118

A dilemma

To obtain the incompleteness

without ω-consistency but only consistency, the key is Rosser’s modification Prf∗

T

for representing the notion “... is a proof of ...”;

but the corresponding consistency statement

Con∗(T) is provable even in Q and hence in T.

– p. 39

SLIDE 119

Loeb’s derivability conditions

A “canonicality” on PvT(u) ≡ ∃xPrfT(x, u): (1) If T ⊢ ϕ then Q ⊢ PvT( ϕ ); (2) IΣ0+Ω1 ⊢ PvT( ϕ→ψ) → (PvT( ϕ) → PvT( ψ)); (3) IΣ0+Ω1 ⊢ PvT( ϕ ) → PvT(PvT( ϕ ) ).

– p. 40

SLIDE 120

Loeb’s derivability conditions

A “canonicality” on PvT(u) ≡ ∃xPrfT(x, u): (1) If T ⊢ ϕ then Q ⊢ PvT( ϕ ); (2) IΣ0+Ω1 ⊢ PvT( ϕ→ψ) → (PvT( ϕ) → PvT( ψ)); (3) IΣ0+Ω1 ⊢ PvT( ϕ ) → PvT(PvT( ϕ ) ). These conditions imply T ⊢ ¬PvT( ⊥ ).

– p. 40

SLIDE 121

Loeb’s derivability conditions

A “canonicality” on PvT(u) ≡ ∃xPrfT(x, u): (1) If T ⊢ ϕ then Q ⊢ PvT( ϕ ); (2) IΣ0+Ω1 ⊢ PvT( ϕ→ψ) → (PvT( ϕ) → PvT( ψ)); (3) IΣ0+Ω1 ⊢ PvT( ϕ ) → PvT(PvT( ϕ ) ). These conditions imply T ⊢ ¬PvT( ⊥ ). (Proof) Self-reference Lemma yields σ s.t. Q ⊢ σ ↔ (PvT( σ) → ⊥).

– p. 40

SLIDE 122

Loeb’s derivability conditions

A “canonicality” on PvT(u) ≡ ∃xPrfT(x, u): (1) If T ⊢ ϕ then Q ⊢ PvT( ϕ ); (2) IΣ0+Ω1 ⊢ PvT( ϕ→ψ) → (PvT( ϕ) → PvT( ψ)); (3) IΣ0+Ω1 ⊢ PvT( ϕ ) → PvT(PvT( ϕ ) ). These conditions imply T ⊢ ¬PvT( ⊥ ). (Proof) Self-reference Lemma yields σ s.t. Q ⊢ σ ↔ (PvT( σ) → ⊥). Then the conditions (1) and (2) yield IΣ0+Ω1 ⊢ PvT( σ) → (PvT(PvT( σ)) → PvT( ⊥ )).

– p. 40

SLIDE 123

Loeb’s derivability conditions

A “canonicality” on PvT(u) ≡ ∃xPrfT(x, u): (1) If T ⊢ ϕ then Q ⊢ PvT( ϕ ); (2) IΣ0+Ω1 ⊢ PvT( ϕ→ψ) → (PvT( ϕ) → PvT( ψ)); (3) IΣ0+Ω1 ⊢ PvT( ϕ ) → PvT(PvT( ϕ ) ). These conditions imply T ⊢ ¬PvT( ⊥ ). (Proof) Self-reference Lemma yields σ s.t. Q ⊢ σ ↔ (PvT( σ) → ⊥). Then the conditions (1) and (2) yield IΣ0+Ω1 ⊢ PvT( σ) → (PvT(PvT( σ)) → PvT( ⊥ )). (3) yields IΣ0+Ω1 ⊢ PvT( σ) → PvT( ⊥ ), and so IΣ0+Ω1 ⊢ ¬PvT( ⊥ ) → σ. Since T ⊢ σ, done!

– p. 40

SLIDE 124

A controversy

In the case of T = Q:

everything must be through Nelson’s

interpretation n of IΣ0+Ω1 in Q;

– p. 41

SLIDE 125

A controversy

In the case of T = Q:

everything must be through Nelson’s

interpretation n of IΣ0+Ω1 in Q;

PvT(u) is (∃xPrf(x, u))n ≡ ∃x(W(x) ∧ Prf(x, u)n);

and Con(T) is ¬∃x(W(x) ∧ Prf(x, ⊥ )n);

– p. 41

SLIDE 126

A controversy

In the case of T = Q:

everything must be through Nelson’s

interpretation n of IΣ0+Ω1 in Q;

PvT(u) is (∃xPrf(x, u))n ≡ ∃x(W(x) ∧ Prf(x, u)n);

and Con(T) is ¬∃x(W(x) ∧ Prf(x, ⊥ )n);

Does this Con(T) really represent “consistency”?

– p. 41

SLIDE 127

A controversy

In the case of T = Q:

everything must be through Nelson’s

interpretation n of IΣ0+Ω1 in Q;

PvT(u) is (∃xPrf(x, u))n ≡ ∃x(W(x) ∧ Prf(x, u)n);

and Con(T) is ¬∃x(W(x) ∧ Prf(x, ⊥ )n);

Does this Con(T) really represent “consistency”?

On the other hand, in the case of T = ZFC,

everything must be through von Neumann’s

interpretation v of IΣ0+Ω1 in ZFC;

– p. 41

SLIDE 128

A controversy

In the case of T = Q:

everything must be through Nelson’s

interpretation n of IΣ0+Ω1 in Q;

PvT(u) is (∃xPrf(x, u))n ≡ ∃x(W(x) ∧ Prf(x, u)n);

and Con(T) is ¬∃x(W(x) ∧ Prf(x, ⊥ )n);

Does this Con(T) really represent “consistency”?

On the other hand, in the case of T = ZFC,

everything must be through von Neumann’s

interpretation v of IΣ0+Ω1 in ZFC;

PvT(u) is (∃xPrf(x, u))v ≡ (∃x ∈ ω)Prf(x, u)v;

and Con(T) is ¬(∃x ∈ ω)Prf(x, ⊥ )v.

– p. 41

SLIDE 129

A controversy

In the case of T = Q:

everything must be through Nelson’s

interpretation n of IΣ0+Ω1 in Q;

PvT(u) is (∃xPrf(x, u))n ≡ ∃x(W(x) ∧ Prf(x, u)n);

and Con(T) is ¬∃x(W(x) ∧ Prf(x, ⊥ )n);

Does this Con(T) really represent “consistency”?

On the other hand, in the case of T = ZFC,

everything must be through von Neumann’s

interpretation v of IΣ0+Ω1 in ZFC;

PvT(u) is (∃xPrf(x, u))v ≡ (∃x ∈ ω)Prf(x, u)v;

and Con(T) is ¬(∃x ∈ ω)Prf(x, ⊥ )v. What’s the difference between them?

– p. 41

SLIDE 130

3. Connection to the present-day

researches

– p. 42

SLIDE 131

Comparison of thoeries

Given S ⊆ T, under which condition, a theory T could be said essentially stronger than another S?

– p. 43

SLIDE 132

Comparison of thoeries

Given S ⊆ T, under which condition, a theory T could be said essentially stronger than another S? (A) there is a sentence ϕ s.t. S ⊢ ϕ and T ⊢ ϕ?

by changing ways of formalizing concepts,

S might be able to simulate T;

e.g., ZFC−FA can simulate ZFC,

and ZFC−Ext can simulate ZFC.

– p. 43

SLIDE 133

Comparison of thoeries

Given S ⊆ T, under which condition, a theory T could be said essentially stronger than another S? (A) there is a sentence ϕ s.t. S ⊢ ϕ and T ⊢ ϕ?

by changing ways of formalizing concepts,

S might be able to simulate T;

e.g., ZFC−FA can simulate ZFC,

and ZFC−Ext can simulate ZFC. (B) there is no interpretation of T in S?

prevents the possibility that S simulates T;

– p. 43

SLIDE 134

Comparison of thoeries

Given S ⊆ T, under which condition, a theory T could be said essentially stronger than another S? (A) there is a sentence ϕ s.t. S ⊢ ϕ and T ⊢ ϕ?

by changing ways of formalizing concepts,

S might be able to simulate T;

e.g., ZFC−FA can simulate ZFC,

and ZFC−Ext can simulate ZFC. (B) there is no interpretation of T in S?

prevents the possibility that S simulates T;

While there is another way to obtain (A), e.g., constructing a model M s.t. M | = S but M | = T,

– p. 43

SLIDE 135

Comparison of thoeries

Given S ⊆ T, under which condition, a theory T could be said essentially stronger than another S? (A) there is a sentence ϕ s.t. S ⊢ ϕ and T ⊢ ϕ?

by changing ways of formalizing concepts,

S might be able to simulate T;

e.g., ZFC−FA can simulate ZFC,

and ZFC−Ext can simulate ZFC. (B) there is no interpretation of T in S?

prevents the possibility that S simulates T;

While there is another way to obtain (A), e.g., constructing a model M s.t. M | = S but M | = T, practically the only way to obtain (B) is showing T ⊢ Con(S).

– p. 43

SLIDE 136

Gödel hierarchy

For theories T and S which are consistent, recursively axiomatizable, essentially containing Q,

S < T iff T ⊢ Con(S);
S ≡ T iff IΣ0+Ω1 ⊢ Con(S) ↔ Con(T).

– p. 44

SLIDE 137

Gödel hierarchy

For theories T and S which are consistent, recursively axiomatizable, essentially containing Q,

S < T iff T ⊢ Con(S);
S ≡ T iff IΣ0+Ω1 ⊢ Con(S) ↔ Con(T).

Large parts of proof theory and set theory are investigations of this hierarchy:

– p. 44

SLIDE 138

Gödel hierarchy

For theories T and S which are consistent, recursively axiomatizable, essentially containing Q,

S < T iff T ⊢ Con(S);
S ≡ T iff IΣ0+Ω1 ⊢ Con(S) ↔ Con(T).

Large parts of proof theory and set theory are investigations of this hierarchy:

measure for <:

proof theoretic ordinal; large cardinal.

– p. 44

SLIDE 139

Gödel hierarchy

For theories T and S which are consistent, recursively axiomatizable, essentially containing Q,

S < T iff T ⊢ Con(S);
S ≡ T iff IΣ0+Ω1 ⊢ Con(S) ↔ Con(T).

Large parts of proof theory and set theory are investigations of this hierarchy:

measure for <:

proof theoretic ordinal; large cardinal.

methods establishing ≡:

cut elimination; forcing; inner model, etc.

– p. 44

SLIDE 140

Picture of the hierarchy

Z2 ≡ ZFC−Pow . . . < Π1

2-CA0

< Σ1

2-AC ≡ KPi ≡ T0

< ID<ω ≡ Π1

1-CA0

< ID1 ≡ BI ≡ KP ≡ CZF ≡ MLT <

ID<ω ≡ IR ≡ ATR0

< PA ≡ ACA0 ≡ Σ1

1-AC0 ≡ HA

. . . < IΣ2 < PRA ≡ IΣ1 ≡ RCA0 ≡ WKL0 < Q ≡ IΣ0+Ω1

– p. 45

SLIDE 141

Picture of the hierarchy

ZFC+Inac Z2 ≡ ZFC−Pow . . . . . . < < ZFC3 Π1

2-CA0

< < MK(:= ZFC2) Σ1

2-AC ≡ KPi ≡ T0

. . . < < ID<ω ≡ Π1

1-CA0

NBG+Π1

1-CA

< < ID1 ≡ BI ≡ KP ≡ CZF ≡ MLT NBG+ETR < <

ID<ω ≡ IR ≡ ATR0

ZF ≡ ZFC ≡ NBG < < PA ≡ ACA0 ≡ Σ1

1-AC0 ≡ HA

Z . . . < < Z<ω ≡ ZBQC IΣ2 . . . < < PRA ≡ IΣ1 ≡ RCA0 ≡ WKL0 Z3 < < Q ≡ IΣ0+Ω1 Z2 ≡ ZFC−Pow

– p. 45

SLIDE 142

Picture of the hierarchy

ZFC+“0 = 1” . . . < ZFC+Vop < ZFC+Inac ZFC+SCpt Z2 ≡ ZFC−Pow . . . < . . . < ZFC+Wood < ZFC3 < Π1

2-CA0

< ZFC+Meas < MK(:= ZFC2) < Σ1

2-AC ≡ KPi ≡ T0

. . . ZFC+0♯ < < < ID<ω ≡ Π1

1-CA0

NBG+Π1

1-CA

ZFC+WCpt < < . . . ID1 ≡ BI ≡ KP ≡ CZF ≡ MLT NBG+ETR < < < ZFC+2-Mahlo

ID<ω ≡ IR ≡ ATR0

ZF ≡ ZFC ≡ NBG < < < ZFC+Mahlo PA ≡ ACA0 ≡ Σ1

1-AC0 ≡ HA

Z . . . . . . < < < Z<ω ≡ ZBQC ZFC+ω-Inac IΣ2 . . . . . . < < < PRA ≡ IΣ1 ≡ RCA0 ≡ WKL0 Z3 ZFC+2-Inac < < < Q ≡ IΣ0+Ω1 Z2 ≡ ZFC−Pow ZFC+Inac

– p. 45

SLIDE 143

Picture of the hierarchy

ZFC+“0 = 1” . . . < ZFC+Vop < ZFC+Inac ZFC+SCpt Z2 ≡ ZFC−Pow . . . < . . . < ZFC+Wood < ZFC3 < Π1

2-CA0

< ZFC+Meas < MK(:= ZFC2) < Σ1

2-AC ≡ KPi ≡ T0

. . . ZFC+0♯ < < < ID<ω ≡ Π1

1-CA0

NBG+Π1

1-CA

ZFC+WCpt < < . . . ID1 ≡ BI ≡ KP ≡ CZF ≡ MLT NBG+ETR < < < ZFC+2-Mahlo

ID<ω ≡ IR ≡ ATR0

ZF ≡ ZFC ≡ NBG < < < ZFC+Mahlo PA ≡ ACA0 ≡ Σ1

1-AC0 ≡ HA

Z . . . . . . < < < Z<ω ≡ ZBQC ZFC+ω-Inac IΣ2 . . . . . . < < < PRA ≡ IΣ1 ≡ RCA0 ≡ WKL0 Z3 ZFC+2-Inac < < < Q ≡ IΣ0+Ω1 Z2 ≡ ZFC−Pow ZFC+Inac

– p. 45

SLIDE 144

Picture of the hierarchy

ZFC+“0 = 1” . . . < ZFC+Vop < ZFC+Inac ZFC+SCpt Z2 ≡ ZFC−Pow . . . < . . . < ZFC+Wood < ZFC3 < Π1

2-CA0

< ZFC+Meas < MK(:= ZFC2) < Σ1

2-AC ≡ KPi ≡ T0

. . . ZFC+0♯ < < < ID<ω ≡ Π1

1-CA0

NBG+Π1

1-CA

ZFC+WCpt < < . . . ID1 ≡ BI ≡ KP ≡ CZF ≡ MLT NBG+ETR < < < ZFC+2-Mahlo

ID<ω ≡ IR ≡ ATR0

ZF ≡ ZFC ≡ NBG < < < ZFC+Mahlo PA ≡ ACA0 ≡ Σ1

1-AC0 ≡ HA

Z . . . . . . < < < Z<ω ≡ ZBQC ZFC+ω-Inac IΣ2 . . . . . . < < < PRA ≡ IΣ1 ≡ RCA0 ≡ WKL0 Z3 ZFC+2-Inac < < < Q ≡ IΣ0+Ω1 Z2 ≡ ZFC−Pow ZFC+Inac

– p. 45