A Formal Proof of Godels Theorem by L. Chwistek

TBz JOURNAL OF STMBouc LOGIC Volume 4, Number 2, June 1939

A FORMAL PROOF OF GODEL'S THEOREM L. CHWISTEK

I. The present paper contains a formal proof of the following theorem of the elementary system > [ 1 0 1 C:1

1 [10] D 1-o(32) ' 1-2 (54) = .0(5) .1(5) [51 F-o (32) = .0 (3).1 (3) [3].

This theorem means that if there were a theorem of -o ( 3 2 ) 0 stating that there is no contradiction in 1-2 (54 ) 2, we would have a contradiction in

2 ( 5 4 ) 2. (This is in accordance with the second theorem of G6del.) Our proof is based on the calculus of the system t= [ 1 0 ] c and of the meta-

system F-o(32)0. Note that we could take'

.8(.0 (E) .1 (.1 (L) ) .0 ( .1 (L ))E) [L ] = Z [.1 (L) .0 (L)] Expr [.1 (L ).O (.1 (L)) [L] Z l= [.1 (L ) .0(L)]

ExprflE.1 (L)L[L]

instead of AxE .1 (L) ( .1 (L) .0 (L ) L) Z. We would then have to do with the simple systems and metasystems of Hetper, which do not contain propositional variables or the logico-semantical axiom.' Our method applies equally to the systems and metasystems of New Foundation and to the simple systems and metasystems of Hetper.

Our proof can be considerably simplified by using a recent result of Hetper concerning ancestral functions.4 Hetper introduces the following abbreviations:

EL XI isasn af 3 [ .0 ( L ) L ] 9L AvaiLn[ = I*4 X [.0 (L)]

{ 2L*SLX I [.0 (L) ]. PrincL ( A ) Z 3 [ .0 ( L) L 1 LA = Z * L .1 ( L) [ .0 ( L)

] A.

DerL (]B) X Y Z 3 [.0 (L) L ]PL L XL gL 9 L A = X*XL PL [.0(L) ] A = Y*L {L [.0 (L) A = Z * L * * (L ) pZ. L [ .0 (L ) ] B.

AnCL (A B) Z 3 [.0 (L) L ]1L 1 lL A {L * Z EIL JlL

[.0() (L)L 2IL D 21 iL v PrincL (A ) ZIL 3 [ .0 ( L) L ]1L lL A EL X1L i1L A (L PIL i1L DerL (B) X1L YLZ1L.

Received December 6, 1938. O Cf. L. Chwistek and W. Hetper, New foundation of formal metamathetmatics, this

JOURNAL, vol. 3 (1938), p. 36. 2 Loc. cit., p. 19. 3 Cf. W. Hetper, Simple systems, metasystems, and generalized metasystems (to appear). 4 Cf. W. Hetper, Relacje ancestralne w systemie semantyki, Archiwum Towarzystwa

Naukoweeo we Lwowie. vol. 9. no. 6. 61

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62 L. CHWISTEK

If A (z1i), B (xi, yc zlc ) are propositions, the expression Anco (A ( o) B (2o go 2o) ) au will be called an ancestral function (we prove without difficulty that this expression is a proposition). The expressions A ( zj ), B (.to go zo ) will be called respectively the principal term and the term of derivation of this function.

Let us now suppose that 4a (a ic) is an ancestral function and A (to), B ( to go io) are its principal term and its term of derivation. In accordance with the theory of Hetper, if

I [10] DA (zl) i (zl), 1 [1 0] DA ' (xi) A ' (yu) B (x1C yIc zI) I (Zr.)

are theorems, then ,[ 1 0 iD 4 (Si.) I' (zi, ) is a theorem. This statement will be called the ancestral pattern. II. Let us begin with some abbreviations which will enable us to get a very

simple symbolic language. We assume the following directions:5

AxE(L) Z n'axebb AxE.2 (L)( *2 (L)1 (L) .0 (L) ) t= 1.2 (L ) .1 (L) 1 Z.

AXL Z " 3 1.0 (L)] 4Ax eL (L ) Z.

DEFGHIK(L)XYZ " DEFGHIK(.2 (L) .1 (L) .0 (L)) I= [.2(L).1 (L)]X t- 1 2 ( L) *1 ( L) ]Y

I= [.2(L).1 (L)]Z. D(L)XYZ DL (.2 (L) )

>= [.2 (L) .1 (L) ]X IL [.2 (L) .1 (L)E Y

|=[.2 (L) .1 (L)]Z. L E It L ( .2 ( L) .1 ( L) ) B.

ImLEF " (E.0 (L).2 (L) F) [L]- SLE " 3 [.0 (L) L ] S2L A Im.o(L) E S2L, L S2L.

SL (AX)L Z 3 [.0 (L) L] 2L A Im.o(L) Z 2L 1- LAX.(L) Z2L.

SL (D)L X YZ 3 [.0 (L ) L ] X2L g2L ZL A Im.o(L) **X Y Z **22L V2L L

F L D.i(L) X2L Y2L Z2L

SLM E It 3 [.0 (L) LO ] S2L A Im.O(L) E S2L HL FMS2L-

Our argument will be based on the following theorem:

6 Cf. New foundation, p. 35.


FORMAL PROOF OF GODEL'S THEOREM 63

1 [10] D FouleSo ule. (a)

We shall prove this theorem using the ancestral pattern. To do it we prove first the following theorem, by a simple application of the principle of induction:

1= [1 0] _ o u1e Anco (Axo (3) Zo Do (3) lo go o) t= [32 uic.

Note that Hetper dispenses with the use of this theorem, taking H L, E as an abbreviation of AncL ( AXL ( .2 (L) ) ZL DL (.2 (L) ) L L ) 21 [.2 (L) .1 (L) ]E.

Now, Ho Z being equivalent to an ancestral function, we can use it in our ancestral pattern. We take in this pattern:

HoZ for 4k(Z),

AxoZ "A (Z),

DoXYZ " B (X YZ),

Sm Z " (Z).

Then we have to prove:

I 0 ] D Axo ule S02 Ulc. (b

1 [10 ] D A S02 xlC So2 yic Do Xic Ylc zicS2c. (C)

To do it, we have first to prove:

= [10] ZAxele (0) zSoAxel (0)zle. (d)

1 [1 0] D Deief1e gic hie iic k1i ( 0) zi y1,c zi, So D efie gic hie ile kle (0) Xic M/c Zie. (e)

We begin with the proof of (c) and of (d).

III. We have by the logical calculus of = [1 0 ] c and by the direction of expressions of Fo 0:

(1) 14[11 0 ] SoExpr [1 ]a, is a theorem.

By the directions of expressions and of pattern of substitution we have:

(2) [1 0]SO Prop ( al, bic1 cl di) 1 ] is a theorem.

Then we get by the direction of propositions:

(3) t- [ I 0 ] SO Prop --- ( alu ble cle di, ) 1 ]is a theorem.

These theorems enable us to prove by the principle of induction:

t [ 1 0 ] D = acb1c [ 1b ] So = ae bl [ 1] is a theorem,

[ 1 0 ] D = au ble (1] So = al. b1 [1 ] is a theorem.

Suppose now we have proved,

= [I0]SoPropE, t= [10]SoPropF;


64 L. CHWISTEK

then we can prove: t= [ S0]S E D vEF, I [l0]So FvEF, = [ S0] DSoESovEF, [10] ZSoFSovEF, t [10] D DESE DESvEF, I [10] :DFSoF ZFSovEF.

Then we have by the logical calculus,

t: [ 10] DA DESE D FSoF DvEFSovEF. Then we get, i [10J DA DESoE DFSoF DAEFSoAEF. Using this method, we get by the principle of induction:

= [10] D (au bl cledi?) [di, SO (ai bsocidcl ) [1d isatheorem, I (1 0] D (ar bu cld1?) [ 1 ] So (au bl, c1,d1) [d 1 ] is a theorem.

Now, using the same method, we prove step by step the theorems (d), (e). Thus we have:

(4) I (10] DAxele(0)z1eSoAxele (0)zleisa theorem, (5) 1 0 D D elfie gi, hie ile kic ( 0) xic yic Z

So D exefii gic hie ilc k1c ( 0) lc yic zic is a theorem. Note that these theorems could not be proved if Ax ele ( ) zle and D elefie gpc hic ilc k1l ( 0 ) xie Yic zle were replaced by arbitrary functions.

Now we pass to the proof of the theorems (b), (c).

IV. Let us assume the following direction: 1 E F G is an abbreviation of D D E F G ha3l i13i k131 (2) xl3i Yiai zif

D2 x131 y?3 zl. We have by the logical calculus of I-o 0: (6) 1=[ 1 0 ] Ho D el3lfi3i g131 is a theorem.

t= [1 0 ] Ho Prop D el3lf131 gi3s is a theorem. 1= [ 1 0 ] Fo Prop Prop D el3l f131 g131 is a theorem.

Now we get by the semantical calculus of 1= [ 1 01 c: (7) 1= (10] D Iml ale ac = ane .0(1) 1 1 is a theorem.

t [1 0] D Iml ale a = alle .0 (2) 1] is a theorem. = (1 0 D Imi ale a { ane I ( 1) [1] is a theorem.

1 0] D Im1 ale an'c an, I ( 2 ) 11] is a theorem. These theorems enable us to prove, step by step, using the semantical calculus of t=[10]c: (8) 1= [ 1 0 1 - I D el3lf131 g13e1s2 1(1 ] is a theorem. Then we get by the logical calculus of Ho 0:


FORMAL PROOF OF GbDEL'S THEOREM 65

(9) 1 [1 0 1 Ho 11 [ 3 2 ] e32 D einf13l 9131 is a theorem. Now we have by the direction of deduction of Fo 0 and by (1) and (6),

1 [ 1 0 ] D Imi eic enc F-o D II [ 3 2] e132 D einfI31 9131 gD elicfi31 g913 is a theorem. Then we get by the direction of modus ponens of Fo 0 in consequence of (9),

(lOa) = [ 1 0 ] D Im eic eilc FHo D elcfi3,9131 is a theorem. The same method enables us to prove,

(lOb) ? [1 0 ] D Imi eic elc F o Prop D ellcfll 9131 is a theorem.

Then we have by the direction of propositions of Fo 0, (lOc) 1= [ 1 0 ] D Imi eic enc F o Prop Prop D ellcfl3l 9131 is a theorem. Now let us remark that the theorems (7) enable us to prove by the elementary semantical calculus:

(11) : [1 0 1 D Im, eic enc 15 elicf131 9g131fi32 } [ 1 ] is a theorem. Then we prove using (l0a), (lOb), (lOc), and (11) instead of (6) and (8),

1= [ 1 0] D A Im ec eil Im, ficfilc Ho 1 elucfulc 91g1 is a theorem. Now we introduce the following directions:

Hip1 is an abbreviation of Imi * * xic Yic zic * * Xiic yuc Zuc.

Hip2 " " " " Im * * * * * eflcgic hicilcklc* * * * * eiicfiic giic hc illc kl.

Hip " " " "A Hip1 Hip2. Using our method of substitution, we get, step by step:

1= [ 1 0 1 D Hip Fo D D enufiic giic hilc ilic klic ( 2 ) xilc ylc Zlic D2 X11c Y11c ZIC is a theorem.

Then we have by the direction of modus ponens of Ho 0, 1 0 1 D Hip D Ho D enc fiic giic hle ilic klic ( 2 ) xiic ylnc Zic

Fo D2 X11C Y1C Z1ic is a theorem.

Now, using the logical calculus of 1 0 ] c, we get:

> [ 1 0 ] D So D e1CfiC gic h1e i1c k1C (0) XiC YiC Zic SO (D)o xic yic zic is a theorem. Then we get by (5),

[ 1 0] D D eicfie gic hic iic kic (0) xic Yic Zic SO (D)o Xic Y1c ziC is a theorem. This theorem enables us to prove by the logical calculus of [1 0 ] c,

(12) [0 1 D Do xic y/c zic SO (D)o xic yic zic is a theorem.

The same method can be applied to Axo Z, yielding:

(13) 1 [1 0 1 D Axo z1c S' (Ax)o z1c is a theorem.


66 L. CHWISTEK

The calculus of I-o 0 enables us to prove:

1= [10] Ho DAxi u21 F2u isiatheorem.

Then we get, by the method of substitution used above, 1= [ 1 0 ] D Im, zi. zu. o HD Ax2 zul F 2 zue is a theorem.

Now, using the direction of modus ponens of Ho 0, we get;

1= [1 0 ] D Im z14 Zu. D Ho Ax2 zue Ho I-a2 z is a theorem.

Then we get by the logical calculus of [ 1 0c,

,t [ 1 0 ] D So (Ax)o zic So0 zc is a theorem.

Then we have by (13),

(14) = [ 1 0 DAxozSzic is a theorem.

Now let us remark that we have by the calculus of Ho 0:

1= [ 1 0 ] Ho D HZ31 2 2 Yai D D2 Xa1 Ya1 Z3i H2 Z31 is a theorem.

Then we have by the method of substitution used above:

1 [1 0 ] D Hip1 -o D H2 Xuc D 1-2 Ync D D2 Xu7c Yllz z11H 2 zll, i8 a theorem.

Applying the direction of modus ponens to F-o 0 three times to this theorem, we get: I: [ 1 0 ] D Hipi D Fo :2 Xa D F-Ho F2n D Ho o F- 2 D2 Xnc yllec HFo F2 ZUc

is a theorem.

Then we get by the calculus of 1= [ 1 0 ] c and by (12),

(15) 1= [ 1 0 ] D A S02ozlc A S02 ylc Do xzc yic zic Sg zici i a theorem.

From (14) and (15) we get by Hetper's method:

(16) 1= [ 1 0] D F-o ul. Soulcis a theorem.

This is our fundamental lemma. In an analogous manner we get, using F-o 0 insteadof 1= [ 1 0 ] c:

(17) 1= [ 1 0 I -o D F-2 U31 S24 U is a theorem.

The theorems (16), (17) enable us to get Godel's theorem by an easy formal derivation.

V. Let us introduce the following directions of abbreviation:

GL ( E ) is an abbreviation of 3 [ .0 ( L ) L ] XL 7L ZL uL A = EI[.2 (L) .1 (L) ]!7LL[.O (L) L

A Im.O(L) E XL A ( ZL 9L XL L) [ .0 (L ) ] I L aL.

G(L) is an abbrevziation of GL ( II [ .2 ( L ) .1 (L) L a.,() G.l(L) ( al()))

Now we have by elementary logical calculus:


FORMAL PROOF OF GODEL'S THEOREM 67

(18) = [ 101 Ho G(2) -- -2 G(4) is a theorem.

(19) [1 1 H]o 1- G(4) -' 1-4 G(e) is a theorem. On the other hand we have by (16) and (17),

(20) t= [ 1 0 1 D Ho G(2) l-o H2G(4) is a theorem,

(21) [ 1 0 ] D Ho -G) 0 l-2 Ha' G(4) is a theorem, and,

(22) I=4101HoDH2G(4)Hs2HG(a) isatheorem,

(23) t= I 01 H]F-H D F2-- G(4) F2 H 4 G(s) is a theorem. From (18) we get by the logical calculus of F0 0:

(24) I=[1 0 ]D Fo G(2)Ho- 2((4) is a theorem. (25) =1 0] D Fo G(2) Fo 2 G(4) is a theorem. Now we get from (20), (24),

(26) = [ 1 0 D Fo G(2) Fo = .0 ( 3 ).1( 3 )3 ]is a theorem. From (21), (25) we get:

1 [ 10] D FH G(2) FHo H2 = .0( 5) .1(5) 51 is a theorem. Therefore, since by (16) and (26),

1= [10] FHoG(2) Ho Ha = .0(5) .1 (5) [5jisatheorem, we have that

1= [10] Dv FoG(2) FHo^.G(2) Fo H2 = .0(5) .1(5) [5]isatheorem.

This theorem corresponds closely to the first theorem of Godel. Notice, how- ever, that there the contradiction does not appear in Fo 0 but in F2 2.

Now we pass to the second theorem of Godel. From the theorems (22) and (19) we get:

t= [ 1 0 ] O o D= 2G(4) H 2 J G(4) is a theorem. Thus we have:

t l 1 0 H FDo G(2) H2 = .0 (5) .1(5) [ 51 is a theorem, = [10] Ho D%0 H2 = .0(5) .1 (5) 15]G(2) isatheorem.

Now, applying modus ponens of Ho 0 and using (26), we get:

1 [1 0]DH -- H 12 = .0 (5) .1 (5) [5] -o=.0(3).1 (3)[3 isathm.

We see that, if we could prove in Fo 0 that there is no contradiction in FH 2, we should have a contradiction in Fo 0. This is G6del's theorem.

Let us remark that there is no way to prove:

If 1 [01 0] ' o H (3 2 ) = .0 (3 ) .1 (3) [3 ] is a theorem, then 1= [1 01 Fo (3 2) . Ha2 (54) = .0 (5).1 (5) [5] is a theorem.


68 L. CHWISTEK

Nevertheless it is obvious that if we could prove,

> [ 1 0 ] H Fo (3 2 ) -.0 (3 ) .1 (3) [3 ] is a theorem, we could prove applying the same method step by step, that

1= [10] -o (32) '_' 1-2 (54) = .0 (5) .1 (5) [5] isa theorem.

Thus there is no hope of proving the consistency of 0o ( 3 2 ) 0.

VI. Summary of the directions concerning the use of variables in the systems and metasystems occurring in this paper:

Systems and Constant Real Apparent Domain of metasystems expressions variables variables substitution

The verbal any E, F, GI H, any expression system expression I, K, L, M, N

The auxiliary any expres- system sion of

type c

au, ble, cli, ao, bo, co) .0 ( 1), ........................... . . .. ....* a ble,

The system any expres- xiv, ylic Zic, xo, 90, yo, * .0 ( 1 ) * a,, bl, 1= [ 1 0 ] c sion of alic, bile, clnn, 410, blo, aio, * * a bl .0 ( 1 ),

type 1 ............... ............ ............... xjle, Ylic, Zilc X2o, Ploy Z10

a3l., b3s, C3l, a2, b2, *, .0 ( 3 ), The any expres- ................ . * a31 b31,

metasystem sion of X31, Y31, Z31I X2, Y2, z * .0 (3 ) * a31 b3l, Ho ( 3 2 ) 0 type 3 a131, b131, c131, dn2, b2,y 12, * * a31 b31 .0 ( 3 ),

. ...... .... .. .. .. .. . .. .. . .. . .. .. .. .. .. .. .. .. ..

X131, Y131, Z131 X12, y12, 12

a63, b63, C53, a4, b4, C4, .0 ( 5 ), The any expres- ........ . .................. * as3 bu3,

metasystem sion of x3, 5ya, ZY, x, 94, Z4, * .0 ( 5 ) * aw bM, 2(5 4 ) 2 type 5 am, b153, C163 d a14, b14,Y l, * * an bu.0 ( 5 ),

.. .. ..... ...... .. . ...... .. .......... .... .. .........

X153, Ylm, Zi X14, Y14, 214

Note that in XKL, ir[, XZKL, and XzL, K and L stand for types, while I is used to build up new letters.6

INSTYTUT LOGMI, LW6W

6 Cf. New foundation, All and p. 35.


Article Contentsp. 61p. 62p. 63p. 64p. 65p. 66p. 67p. 68

Issue Table of ContentsThe Journal of Symbolic Logic, Vol. 4, No. 2 (Jun., 1939), pp. 41-104Symbols, Signs, and Signals [pp. 41-52]An Informal Exposition of Proofs of Gdel's Theorems and Church's Theorem[pp. 53-60]A Formal Proof of Gdel's Theorem[pp. 61-68]On the Independence of the Axioms of Definiteness (Axiome Der Bestimmtheit) [pp. 69-72]A System of Strict Implication [pp. 73-76]A Theorem on Deducibility for Second-Order Functions [pp. 77-79]Definition by Induction in Quine's New Foundations for Mathematical Logic [pp. 80-81]ReviewsReview: untitled [pp. 82-87]Review: untitled [pp. 87-88]Review: untitled [pp. 88-89]Review: untitled [pp. 89-90]Review: untitled [pp. 90]Review: untitled [pp. 90-91]Review: untitled [pp. 91]Review: untitled [pp. 91]Review: untitled [pp. 91-92]Review: untitled [pp. 92]Review: untitled [pp. 93]Review: untitled [pp. 93-94]Review: untitled [pp. 94-96]Review: untitled [pp. 96-97]Review: untitled [pp. 97-98]Review: untitled [pp. 98]Review: untitled [pp. 98-99]Review: untitled [pp. 99-100]Review: untitled [pp. 100]Review: untitled [pp. 100]Review: untitled [pp. 101]Review: untitled [pp. 101]Review: untitled [pp. 102]Review: untitled [pp. 102]Review: untitled [pp. 102]Review: untitled [pp. 102]Review: untitled [pp. 103]Review: untitled [pp. 103]Review: untitled [pp. 103]Further Citations [pp. 103-104]

A Formal Proof of Godels Theorem by L. Chwistek

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