notes on smith's an introduction to godels incompleteness theorems

7/31/2019 Notes on Smith's an Introduction to Godels Incompleteness Theorems

1/42

N : 0, 1, 2, . . .

+,

T

T T

T N

T T

T

T

T T


2/42

T GTT T

T

T GTGT

GT T

GT T

T

T GT GT T

T GT T

GT

GT T GT

T GT GT

T T

{T + GT} S GS

, {}, {, {}}, . . .

T

0 = 1

ConT T 0 = 1

T GT ConT GT

T ConT GT T GT T ConT


3/42

f : N N

{x|f(x) } = {f(x) : x }

f x x f(x) = y

f f(x) = f(y) x = y

f

P cp cp(x) =0 P(x) cp(x) = 1 P(x)

1

01


4/42

f : N f

P NX P X N

f P f : N P

D N n D n / f(n)

D P f Pd f(d) = D

n n f(d) n / f(n) d f(d) d / f(d)

f PP 2

n, m

f : N N2 n n 2

f(n, m) N f(n, m) = (n+m)(n+m+1)2 + m


5/42

f :

x f(x)

N

N

2

c c2

Nf

f : N

c

c(x) = {0 : x

f : N

f(0) 0 f(0) 1f(1), f(2), . . .

f : N 2


6/42


7/42

e K e / We K = We e K0 1 . . . K

K K K

L L,I LI L

L L

LL

L

L L

I L L

T T

T T

T T


8/42

T T T

T T T

T T

T T

T T T

T TT T T

TT

TT

T2

2

T 2

T

T T

sans serif

LALA = LA,IA

LA


9/42

{0, S, +, } 0 S+

LA

LA

IA N

IA

0 val[0] = 0

val[S] = val[] + 1

val[( + )] = val[] + val[] val[( )] =val[] val[]

LA =

P (x)L n

n P (n)n P (n)

P

T P (x) n

n P T (n)n P T (n)

P

T (x) P T (x) P

L

L L


10/42

K Kf K n K x f(x) = n x

f LL f

F(x, y) f(m) = n F(m, n)

n K x(Nat(x) F(x, n))n K x(Nat(x) F(x, n))

L T

F T x(Nat(x) F(x, n))n n K

K T 2

T T

TL

T

T T

T T TT

TT T T

T T 2


11/42

T

T (x) T (x)

T TD

n D T n(n)

d(x) D TT

d(x) T

(x) T

d(x) T

D d(x) d(x) D n

n D T d(n)n T d(n)

n = d d D T d(d) d D T d(d)

T T 2

T (1 & 2 &3 4)

T (1 & 5 & 2 4)

(1 & 5 & 2 3) 2

BA

BABA LB LB = LB,IB LB

IB


12/42

=

BA

0 = S

S = S =

+ 0 =

+ S = S(+ )

0 = 0

S = ( ) +

BA

BA

= BA =

= BA =

j + k = m j k = m

2

m nm = n = m, = n

BA m = n. 2

BA 2

Q

Q BA Q

BA

x(0 = Sx)

xy(Sx = Sy x = y)

x(x = 0 y(x = Sy))

x(x + 0 = x)

xy(x + Sy = S(x + y))


13/42

x(x 0 = 0)

xy(x Sy = (x y) + x)

Q

Q n Q 0 + n = n

x(0 + x = x)Q 2

Q

Q

Q

v(v + x = y) Qm, n m n Q v(v + m = n) Q v(v + m = n)

m n k 0 k +m = n Q k + m = nQ v(v + m = n)

m n Q v(v + m = n)

LA Q v(v + = )

( )() ( )()( ()) ( ())

T

T x(0 x)

n T x({x = 0 x = 1 . . . x = n} x n)

n T x(x n {x = 0 x = 1 . . . x = n})

n T (0), T (1), . . . , T (n) T (x n)(x)

n T (0) T (1), . . . , Q (n) Q (x n)(x)

n T x(x n x Sn)

n T x(n x (n = x Sn x))

n T x(x n n x)

n > 0 T (x n 1)(x) (x n)(x = n (x))


14/42

Q

Q

T x(0 x)

Q x(x + 0 = x)

Q xz(z + 0 = x)

Q x(0 x) 2

n Q x({x = 0 x = 1 . . . x = n} x n)

a = 0 a = 1 . . . a = na n a n

a (a = 0 a = 1 . . . a = n) nT x({x = 0 x = 1 . . . x = n} x n) 2

n Q x(x n {x = 0 x = 1 . . . x = n})

x(x n {x = 0 x = 1 . . . n = 0 n = k

n = k + 1n = 0

Q a 0 a = 0

Q a = 0 a = Sa

Q b + a = 0

Q b + Sa = 0

Q S(b + a) = 0

Q a 0 a = 0 2n T (0), T (1), . . . , T (n) T (x n)(x)

Q (0) (1) . . . (k)

Q x(x k)(x)

Q ((0) (1) (k))

Q (0) (1) . . . (k) x n)(x) 2

n Q (0) Q (1), . . . , Q (n) Q (x n)(x)

Q (0) Q (1), . . . , Q (n)

k n Q (k)Q x n(x)

n Q (0) Q (1), . . . , Q (n) 2Q (x n)(x)

n Q x(x n x Sn)

Q a n

Q a = 1 . . . a = n


15/42

Q a = 1 . . . a = n a = Sn

Q a Sn

Q x(x n x Sn)

n Q x(n x (n = x Sn x))

Q n a

Q n = a

Q n = 0 . . . n = a

Q n = . . . n = a 1

Q n = . . . Sn = a

Q n a n = a Sn a

Q (Sn a)

Q n = 0 . . . n = a

Q (Sn = 0 . . . Sn = a)

Q n = a

Q n a Sn a n = a

Q n a (n = a Sn a)

Q x(n x (n = x Sn x)) 2

n Q x(x n n x)

n = 0n = 0 0 a

0 a a 0n = k

n = k + 1

Q a k k a

Q a k a k + 1

Q k a (k = a k + 1 a)

Q a = k a k + 1

Q a k + 1 k + 1 a

Q x((x k k x) (x k + 1 k + 1 x)) 2

n > 0 Q (x n 1)(x) (x n)(x = n (x))

Q (x n 1)(x)

Q (0) . . . (n 1)

Q (x n)(x = n (x))

Q (x n)(x = n (x)

Q (0 = n (0)) . . . (n = n (n))

Q ((0 = n (0)) . . . (n = n (n)))

Q (x n 1)(x) (x n)(x = n (x)) 2

0 = ,


16/42

0

0

0 ( ) ( ) ( ) ( )

0 ( ) ( )

1 . . . 0 , , . . . ,

1 1

1 . . . 0 , , . . . ,

1 1

0 0

2

1 1 1 1

2

0 1 1

0 ( z = z)0 z( z = z) 1

11

0

T T

T T

Q 1

Q 1


17/42

Q 0

Q 0

Q 1

1 Q Q

1 QQ 1 Q 1

1 Q 1 1 Q 2

L2 L1

L1 L2 L2

L1 L2

T2 T1 T2 T1 T2T1

T Q T 1

T 1 1 T

T 1 1T Q

1 T

T 1 1

T T 1T 1 2

P A QP A

P


18/42

Pn n + 1

P A

P A

({0 x((x) (Sx))} x(x))

Q0

x(x = Sx)

(x) = (x = Sx)

(0)

x(0 = Sx)

0 = S0 2

(n) (Sn)

n = S(n)

xy(x = y Sx = Sy)

n = Sn Sn = SSnSn = SSn 2

x(0 + x = x)

(x) = (0 + x = x)

(0)

x(x + 0 = x)

0 + 0 = 0 2

(n) (Sn)

0 + n = nxy(x + Sy = S(x + y))

0 + Sn = S(0 + n)

0 + Sn = Sn

xy(Sx + y = S(x + y))

(x) = Sa + x = S(a + x)


19/42

(0)

S(a + 0) = S(a + 0)

S(a) = S(a + 0)

Sa + 0 = S(a + 0) 2

(n) (Sn)

Sa + n = S(a + n)

Sa + n = a + Sn

S(Sa + n) = S(a + Sn)

Sa + Sn = S(a + Sn) 2

xy(x + y = y + x)

(x) = x + a = a + x

(0)

0 + a = a + 0

0 + a = a

0 + a = a + 0 2

(n) (Sn)

n + a = a + n

S(n + a) = S(a + n)

Sn + a = S(a + n)

Sn + a = a + Sn 2

xyz(x + (y + z) = (x + y) + z)

(x) = x + (a + b) = (x + a) + b

(0)

a + b = a + b

0 + (a + b) = a + b

0 + (a + b) = (0 + a) + b 2

(n) (Sn)

n + (a + b) = (n + a) + b

S(n + (a + b)) = S((n + a) + b)

Sn + (a + b) = S((n + a) + b)

Sn + (a + b) = S(n + a) + bSn + (a + b) = (Sn + a) + b 2

xyz(x + y = x + z y = z)

(x) = x + a = x + b a = b

(0)

0 + a = 0 + b


20/42

a = 0 + b

a = b

0 + a = 0 + b a = b 2

(n) (Sn)

n + a = n + b a = b

Sn + a = Sn + b

S(n + a) = S(n + b)

n + a = n + b

a = b

Sn + a = Sn + b a = b 2

xy(x y y x)

(x) = x a a x

(0)

0 a a 0v(v + 0 = a)

v = a. 2

(n) (Sn)

n a a n

n a c c + n = aS(c + n) = Sa c + Sn = Sa

c = Sc c = 0Sc + Sn = Sa S(c + Sn) = Sa

c

+ Sn = a c = 0 0 + Sn = SaSn = Sa n a Sn a a Sn

c c + a = nS(c + a) = Sn Sc + a = Sn

a n Sn a a Sn2

xy((x y y x) x = y)

(x) = x a a x x = a

(0)

0 a a 0

a = 0

a = Sa

a 0

Sa 0

Sa = 0


21/42

a = 0

0 a a 0 a = 0 2

(n) (Sn)

Sn a a Sn

(Sn = 0 Sn = 1 . . . Sn = a) (a = 0 a = 1 . . . a = Sn)

Sn = a

Sn a a Sn Sn = a 2

xyz((x y y z) x z)

(x) = (x a a b) x b

(0)

0 = 0

0 = 0 0 = 1 . . . 0 = b

0 b 2

(n) (Sn)

Sn a a b

Sn = 0 Sn = 1 . . . Sn = a

a = b

Sn b

a < b

Sn = 0 Sn = 1 . . . Sn = a . . . Sn = b 2

Q 0 1n > 1 1

xy(x, y)

w(x w)(y w)(x, y)

a, b, (a, b) a b b a(a b b b (a, b)) (b a a a (a, b

2

P A

LAQ

P A

x(0 = Sx)


22/42

xy(Sx = Sy x = y)

x(x = 0 y(x = Sy))

x(x + 0 = x)

xy(x + Sy = S(x + y))

x(x 0 = 0)xy(x Sy = (x y) + x)

({(0) x((x) (Sx))} x(x))

LP P A

PP

BA PP A

n + 1

x + 0 = xx + Sy = S(x + y)

x 0 = 0x Sy = (x y) + x

P(0)P(Sx) P(x)

f(0) = gf(Sy) = h(y, f(y))


23/42

f(x, 0) = g(x)f(x,Sy) = h(x,y ,f (x, y))

S Z(x) =0 k Iki (x1, x2, . . . , xk) = xi k i, 1 i k

f(x , 0) = g(x )f(x ,Sy) = h(x , y, f (x , y))

g(y ) h(x ,u, z ) x z ff(x , y , z ) = h(x , g(y ), z )

n f(n) = g(n)

. . .

f0 f0(0) f0(1) f0(2) f0(3) . . .f1 f1(0) f1(1) f1(2) f1(3) . . .

f2 f2(0) f2(1) f2(2) f2(3) . . .

f3 f3(0) f3(1) f3(2) f3(3) . . .

(n) = fn(n) + 1 nn

1

2

PcP m P cP(m) = 0 m P cP(m) = 1

R cRm R n cR(m, n) = 0 m R n cR(m, n) = 1


24/42

x x. . .

f(x ) n f(x ) = yn + 1

ff(x) = y c(x, y) = sg(|f(x) y|)

c n

2

p(x) Psg(p(x)) P sg

sg(p(x)) PP

p(x) q(x)P Q p(n) q(n) 0p(n) q(n) P Q

P Q

2

2

P f(n) = (x n)P(x)g(n) P f(n) = (x g(n))P(x)

p P k

f(0) = 0f(n) = k(n 1) + k(n 2) + . . . + k(1) + k(0) n > 0


25/42

k(i) = 1 i P k(i) = 0 j Pf(n) = (x n)P(x)

2

f(n) = sg(c0(n))f0(n)+sg(c1(n))f1(n)+. . .+sg(ck(n))fk(n)+c0(n)c1(n) . . . ck(n)a

sg(ci(n)) = 1 Ci(n)ci(n) = 1 Ci(n) 2

m = n m < n m n

m = n sg(|m n|)

m < n m n sg(monus(Sm,n) sg(monus(m, n))2

m|n m n

m|n (y n)(0 y 0 < m m y = n)

2

Prime(n) n Prime

Prime(n) n = 1 (u n)(v n)(x v = n (u = 1 v = 1))

2

0, 1, 2, . . . (n) n

n n

0 = 2Sn = (x n! + 1)(n < x Prime(x))

2

exp(n, i) in exp


26/42

exp(n, i) = (x n){(xi |n) (x+1i |n)}

2

len(0) = len(1) = 0 n > 1 len(n) nn len

p(m, n) = sg(pf(m, n))

p(m, n) = 1 m n

len(n) = p(0, n) + p(1, n) + . . . + p(n 1, n) + p(n, n)

len

l(x, 0) = p(0, x)l(x,Sy) = (p(Sy,x) + l(x, y))

len(n) = l(n, n)2

(Prime(m) m|n) m npf(m, n)

QP A

(x, y)L m, n

f(m) = n (m, n)f(m) = n (m, n)

(x, y) Tm, n

f(m) = n T (m, n)f(m) = n T (m, n)

f (m, n) f

f (m, n) T

m T !y(m, y)


27/42

m,n,

f(m) = n T y(m, n)

f(m) = n T y(m, n)

f (x, y) T

T x!y(m, y)

m,n,

f(m) = n T y(m, n)

f(m) = n T y(m, n)

f T f

f T f

f T fT T Q

f T Q

(x, y) (z y)((x, z) z = y)

m m, x) nf

f T f T

(x, y) =df {(x, y) !y(x, y)} {y = 0 !y(x, y)}

Q

T T

Q

f 0 0

f 0 0

f 0 0


28/42


29/42

Q

Atom,Wff,Sent, P rf

= ( ) 0 S + x y z . . .

. . .

e k + 1 s0, s1, s2, . . . , sk eci si

ci i + 1 i

c00 c11 c22 . . . ckk

P rf

P rf(m, n) m P An

Term(n) n LA

Atom(n) n LA

W f f(n) n LA

Sent(n) n LA

Term(n) Atom(n) W f f(n) Sent(n) P rf(m, n)


30/42

y(y = )

diag(n) n

exp(n, i) n

len(n) n

m nm

n

num(n) n

Sn n > 0 Sn

num(0) = 0 = 221

num(Sx) = S num(x) = 223 num(x)

num 2

diag(n)

V ar(n) n

Termseq(m, n)

Term

Atom(n) n

P A

Gdl(m, n) m P An Gdl(m, n)


31/42

Gdl(m, n) P rf(m, diag(n)) GdlP rf diag

Gdl(m, n) 2

Gdl(x, y) 1 Gdl

U(y) =def xGdl(x, y)U

G =def y(y = U U(y)) U

G xGdl(x, U)

G P A

G m Gdl(mU) Gm m P A U

U G G

m P A G GG P A 2

P A LA P A P A

P A G P AP A GG G P A

G

P AP A

G

1

Gdl(x, y) 1 Gdl(x, U) 11 G 1 xGdl(x, U) 2

1 0


32/42

G 1

P A LA P A P A

Gdl Gdl

P A P A G

G P A mG U

Gdl(m, U) Gdl Gdl P A Gdl(m, U) G P A xGdl(xU)

G P A Gdl(m, U)P A G P A 2

P A

P A P A G

G P A GG U Gdl(m, U) Gdl

Gdl Gdl(m, U) P A Gdl(m, U)G xGdl(x, U)

2

T (x) T(m) x(x)

T

P A

P A P A G P A xGdl(x, U) G

U m

Gdl(m, U) m P A Gdl(m, U)P A 2

T (x) T(m) T x(x)

T (x) T x(x) m T (m)


33/42

T T

P A P A G

P A P A G P A xGdl(x, U)

Gdl(m, U) m P A Gdl(m, U)m P A

G 2

LA P AP A P A P A

P AP A

TT T

T

T LALA T T

T T GT TT+ = {T + GT} T

+

G

P A GPA+G P A + G

T Q

T LA T T T


34/42

QQ

T LALA T T T

T

T LALA T

T T

LA

P f rt(m, n) GdlT(m, n)

GdlT LA GT

GT TT GT T GT

T Q TT P f rt(m, n) GdlT(m, n)

GT

T T GT T T GT

T T

GT T GTT

T Q + CC QC

GC

LA

GT TT + GT T + GT

GT


35/42

T

TT

TT T j T

(j j . . . j) n n = s(j) T

T T T T

T

T 2

T LALA T T

T Q LA T T T

T

TA

T TAT LA

LA T T T TA 2


36/42

1 1

TT RT

T

T LA T T

T RTRT RT

RT RT2

T (x) T x(x) m

T (m)

T 0 (x) T x(x)m T (m)

T 1 1 T

T 1

T T1 T 1 2

T T 1

T 1 LA T

T LA T

Prf(x, y) 1 P rf

Prof(x) =def v Prf(v, x)

T

T T ProvT() T T ProvT() T


37/42

T P rfT(m, )

PrfT P rfT PrfT(m, )

T ProvT(). 2

T T ProvT()T T PrfT(m, )

T 2

P A GP A

G P AG

G Prov(G)

G =df y(y = U U)

U =df xGdl(x, y)

Gdl Gdl

Gdl(m, n) =df P rf(m, diag(n))

Gdl(x, y) =df z(Pfr(x, z) Diag(y, z))

U =df xGdl(x, y)

U =df xz(Prf(x, z) Diag(y, z)) Gdl

U zx(Prf(x, z) Diag(y, z))

U z(Diag(y, z) xPrf(x, z))

U =df z(Diag(y, z) Prof(z)) Prov

U =df U

PA G Prov(G)


38/42

PA G y(u = (U U)

PA G U(U)PA G z(Diag(U, z) Prov(z))

PA G z(z = G Prov(z))PA G Prov(G) 2

T T GT ProvT(GT)

T (x) T T

()

(y) =df z(DiagT(y, z) (z))DiagT T

(y)() diagT() = T z(DiagT(, z) z = ) T z(DiagT(, z) (z)) z(z = (z))

T (). 2

T (x) 11 (x)

diagT1 (y) =df z(DiagT(y, z) (z)) diagT

() 2

T (x)(x)

T DiagT(x, y) diagT Diag

T(x, y) =df [DiagT(x, y) ]

2


39/42

T ProvT(x) T T T

2

P rfT(m, n) m T

n P rfT(m, n)T PrfT(x, y)

RProvT(x) =df v(PrfT(v, x) (w v)PrfT(w, x))

RTRProvT(x) T T RT RProvT(RT)

T RProvT(x) T T

T ProvT

True

True(n) n L

L T(x) L L T()

T L L T(x) T T() L

T L T

T(x) T 2

LTrueL


40/42

LA L2AL2A

X, Y, Z, . . .

X({X0 x(Xx XSx)} xXx)

Xx(Xx (x))

L2A

Q

P A2

L2A

P A2 LA P A P A

P A2 P A P A

T L J TL I

T J T JT

P A P A2

I2A P A2 P A2 L2A

P A2


41/42

P A2 G2 P A2 |=2 G2 P A2

P A2P(M)

M = (M, Sm, 0, s, +, )

ACA0 QACA0 N

ACA0 P A ACA0 P A

P A ACA0

M = (M, Sm, 0, s, +, )Sm M

M |= ACA

0P A P A M M |= ACA0 +

11 CA 11

11 CA ACA0

RCA0 01

01 : n((n)

(n)) Xn(Xn (n))RCA0 Sm = {X|X } RCA0

I1RCA0

RCA0 RCA0

RCA0

RCA0 [0, 1]

RCA0 [0, 1]

RCA0

11 CA

AT R P AACA0 P A

W KL0 I1RCA0 I1


42/42

BW

RCA0

RCA0 ACA0 BW

RCA0 ACA0 KL

RCA0 ACA0

RCA0 ACA0 [N]3

RCA0 W KL0

RCA0 W KL0 [0, 1]

RCA0 W KL0

RCA0 AT R0

Prov() P A

Con =df Prov()

Con 1

P A P A G

P A Con Prov(G)

P A

P A G Prov(G)

P A P A Con

T P A T ConPA

T P A P A ConT

P A P A Con

notes on smith's an introduction to godels incompleteness theorems

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