Gödel s Proof. Henrik Jeldtoft Jensen Dept. of Mathematics Imperial College. Kurt Gödel

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1 Gödel s Proof Henrik Jeldtoft Jensen Dept. of Mathematics Imperial College Kurt Gödel

2 ON FORMALLY UNDECIDABLE PROPOSITIONS OF PRINCIPIA MATHEMATICA AND RELATED SYSTEMS 11 by Kurt Gödel, Vienna 1 The development of mathematics in the direction of greater exactness has as is well known led to large tracts of it becoming formalized, so that proofs can be carried out according to a few mechanical rules. The most comprehensive formal systems yet set up are, on the one hand, the system of Principia Mathematica (PM)2 and, on the other, the axiom system for set theory of Zermelo-Fraenkel (later extended by J. v. Neumann).3 These two systems are so extensive that all methods of proof used in mathematics 27. today have been formalized in them, i.e. reduced to a few axioms and Su rules x(n y) of inference. º ez {z [Pr(l(x)+l(y))]x+y It may therefore be & surmised [($u,v)u,v that x these & x = axioms u * R(b and Gl x) rules * v of & inference z = u * y are * v & also n = sufficient l(u)+1]} to decide all mathematical questions which can in any way at all be expressed formally in the systems concerned. It is shown below that this Su is x(n y) not the derives case, and from that x on in both substituting the systems y in place mentioned of the n-th there term are of in fact x (it relatively being assumed simple problems that 0 < n in the l(x)). theory of ordinary whole numbers4 which 28. [174] 0 cannot St v,x be º en decided {n l(x) from & the v Fr axioms. n,x & not This ($p)[n situation < p is l(x) not & due v Fr in some p,x]} way to the special (k+1) St nature v,x º of en the {n systems < k St v,x set & up, v Fr but n,x holds & ($p)[n for a < very p < extensive k St v,x & class v Fr of p,x]} formal 2 systems, including, in particular, all those arising from

3 The essence First theorem of undecidability: If axiomatic theory is consistent, there exist theorems which can neither be proved or disproved 3

4 The essence Second theorem of undecidability: There is no constructive procedure which will prove axiomatic theory to be consistent. 4

5 Euclid s Elements lived circa 300 BC 23 definitions 5 postulates 465 propositions 5

6 Euclid s Elements The axioms It is possible to draw a straight line from any point to any point. to produce a finite straight line continuously in a straight line. to describe a circle with any centre and radius. That all right angles equal one another. Parallel lines don t cross 6

7 Euclid s Elements Consistency Can mutually inconsistent statements be derive from a set of axioms. Say in Euclid s geometry 7

8 Euclid s Elements In other words Can we be sure no one some day derives a proposition which contradicts another proposition. 8

9 PM PM 9

10 Undecidable Russell s s paradox: Two types of sets: Normal those who don t t contain themselves: A A & Non-normal those who do contain themselves B B 10

11 Undecidable Examples: Normal set: A = the set of MPhys 2 student A A Non-normal: B = the set of all thinkable things B B 11

12 Undecidable Define: N = Set of all Normal sets Question: Is N normal? Assume N is Normal then N is member of itself, since N contains all Normal Sets per its definition i.e., N N. But if N N then N is non-normal So N being Normal implies N being non-normal! 12

13 Undecidable Define: N = Set of all Normal sets Assume N is non-normal then N is member of itself per definition of non-normal. But if N is non-normal it is a member of itself, and N contains per definition Normal sets, i.e., N is Normal. So N being non-normal implies N being Normal! 13

14 The problem is Self-reference 14

15 How to determine the truth of: I I am a liar! Epimenides paradox 15

16 The strategy of Gödel s proof Distinguish between: & mathematics x = x, x 2 = 9 meta-mathematics x=4 is not a solution of x+2=3, PM is consistent 16

17 The strategy of Gödel s proof Enumeration of formalised system: Signs: 1 not = 0 s Gödel number 2 or 3 If then 4 there is an 5 equals 6 zero 7 immediate successor 17

18 The strategy of Gödel s proof Enumeration of formalised system: Math formulas: Gödel number ( x )( x = sy) L There is a number x following right after y 18

19 The strategy of Gödel s proof Enumeration of formalised system: Meta-maths: The formula G is not demonstrable using the rules of PM Gödel number 19

20 The crunch Gödel constructed a formula G for which he showed that: G is demonstrable non G is demonstrable 20

21 More crunch The meta-mathematical content of G is: G= The formula G is not demonstrable Or formally within PM: ( x)dem( x,sub(n,17,n)) 21

22 And more crunch G= The formula G is not demonstrable So since G cannot be demonstrated, it is, per definition, TRUE, though its TRUTH cannot be proved within PM 22

23 The Conclusion All axiomatic systems will contain true propositions which cannot be proved within the system And contain propositions which cannot be determined whether true of false 23

24 Some consequences: The continuum hypothesis: Cantor No set can have a number of elements between and the cardinality of the natural numbers the cardinality of the real numbers 24

25 Cardinality: The real numbers cannot be countered Cantor Proof: assume the opposite r M 1 r r 2 3 = = = 0. x 0. x 0. x x x x x x 13 x L L L r = 0. x x x 11 r = ~ x ~ x ~ x

26 Cardinality: So clearly: # reals > # integers Cantor But: is there a set with a number of elements in between? Cantor said: No - but could not prove it. 26

27 Consequences continued: The continuum hypothesis: Was the first problem in Hilbert s list of 23 unsolved important problems. Paul Cohen showed in 1963 that the continuum hypothesis is undecidable 27

28 More consequences: Truth cannot be identified with provability Roger Penrose: Creative mathematicians do not think in a mechanistic way. They often have a kind of insight into the Platonic realm which exists independently of us. 28

29 More consequences: We cannot build one all embracing explanation of everything based on one finite set of axioms. There exist more true statements than the countable number of truths that can be recursively deduced from a finite set of axioms. 29

30 The world is too complex for a `finitistic axiomatic approach to suffice. Creativity is needed at all levels of description. 30

31 So where does this leave The Theory of Everything? 31

Victoria Gitman and Thomas Johnstone. New York City College of Technology, CUNY

Gödel s Proof Victoria Gitman and Thomas Johnstone New York City College of Technology, CUNY vgitman@nylogic.org http://websupport1.citytech.cuny.edu/faculty/vgitman tjohnstone@citytech.cuny.edu March