Gödel and Incompleteness Tom Cuchta – p. 1/1
Introduction In ancient Greece, Euclid pioneered the geometrical axiomatic system. – p. 2/1
Axiomatic Geometry 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended inde fi nitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. Given any straight line and a point not on it, there exists one and only one straight line which passes through that point and never intersects the fi rst line. – p. 3/1
New Geometries In the mid-19th century, Russian mathematician Lobachevsky and Hungarian Bolyai are credited with exploring mathematics changing Euclid’s parallel postulate. – p. 4/1
Formalism In the early 20th century, formalism was a gaining popular philosophy in mathematics championed by David Hilbert which wanted to prove all of mathematics using formal systems theory. – p. 5/1
Principia Mathematica Formalism culminated in Principia Mathematica by Bertrand Russel and Alfred Whitehead. – p. 6/1
Kurt Gödel Kurt Gödel was a logician from Austria, who in the early 1930’s proved that the formalists’ quest of "if it is true, we will prove it" to be an un fi nishable goal. He did this by adapting a paradox similar to the ancient Greek paradox "This sentence is not true." – p. 7/1
Formal Systems A formal system is a system, L = ( A , R , W , I ) , where A - symbols, R - strict rules between the symbols, W - well formed formulas (or wffs) derived by the rules, and I is an interpretation for all these things. With an initial set W ax ⊂ W , we can generate the rest of W using the symbols in A and rules of R . – p. 8/1
MIU-System A = { M, I, U } W ax = { MI } R MIU = { if MyI ∈ W MIU → MyIU ∈ W MIU ; if My ∈ W MIU → Myy ∈ W MIU ; if MyIIIz ∈ W MIU → MyUz ∈ W MIU ; if MyUUz ∈ W MIU → Myz ∈ W MIU } I is empty – p. 9/1
Peano Arithmetic Let Peano Arithmetic, abbreviated PA, be a formal system with A PA = {¬ , ∧ , ∨ , → , ↔ , ∀ , ∃ , = , ( , ) , 0 , S, + , · , x, y, z, . . . } , where x, y, z, . . . represent the in fi nitude of possible variables in this system. We will let I de fi ne these symbols using their standard number theory de fi nitions. For example, ∧ means "logical and", S is a function that means "successor of", and ∀ is a quanti fi er that means "for all". And R is assigned according to standard fi rst order logic. – p. 10/1
Peano Arithmetic Peano Arithmetic is a formal system created to formalize arithmetic with the following familiar axioms: ∀ x (0 � = Sx ) Property of 0 ∀ x ∀ y ( Sx = Sy → x = y ) Property of S ∀ x ( x � = 0 → ∃ y ( x = Sy ) Property of S ∀ x ( x + 0 = x ) Property of 0 ∀ x ∀ y ( x + Sy = S ( x + y )) Property of + ∀ x ( x · 0 = 0) Property of 0 ∀ x ∀ y ( x · Sy = ( x · y ) + x ) Property of · ( { φ (0) ∧ ∀ x ( φ ( x ) → φ ( Sx )) } →∀ xφ ( x ) Induction Schema – p. 11/1
Gödel Numbering De fi ne the code-numbers of each symbol in PA as the natural number below it in the following table: = ( ) 0 � ∧ ∨ → ↔ ∀ ∃ S 1 3 5 7 9 11 13 15 17 19 21 23 + . . . · x y z 25 27 2 4 6 . . . Let expression e be a sequence of k + 1 symbols s 0 , s 1 , . . . , s k . Then e ’s Gödel Number is calculated by taking the code-number c i for each s i , using c i as an exponent for the i + 1 -th prime number, and the multiplying the results to the natural number � e � = 2 c 0 · 3 c 1 · . . . · ( nth prime number ) c n . – p. 12/1
Example of Gödel Numbering Consider the PA-wff ∃ y ( S 0 + y ) = SS 0 . We can talk about its Gödel number by computing it according to the de fi nition. This yields the Gödel number of ∃ y ( S 0 + y ) = SS 0 to be constructed by the following table: y ( S 0 + y ) = S S 0 ∃ 13 4 17 23 21 25 4 19 15 23 23 21 2 13 3 4 5 17 7 23 11 21 13 25 17 4 19 19 23 15 29 23 31 23 37 21 – p. 13/1
A Spin on Gödel Numbers Let e 0 , e 1 , . . . , e n be a sequence of wffs. By de fi nition, each wff corresponds to a particular Gödel number, call them g 0 , g 1 , . . . , g n . Construct a super Gödel number by repeating the process of Gödel numbering on this new sequence, yielding speci fi cally the number 2 g 0 · 3 g 1 · 5 g 3 · . . . · ( nth prime number ) g n . – p. 14/1
Proofs, Consistency and Com- pleteness A proof of a wff φ is a sequence g 1 , g 2 , . . . , g n such that g 1 is known to be true and each g i follows logically from g i − 1 and φ follows from g n . A formal system L is called consistent if no contradictions are derived in the formal system. A formal system L is called complete if for all legal sentences φ , either φ or ¬ φ is derivable through L ’s theorems. – p. 15/1
The Result Gdl ( m, n ) is a function that holds just if m is the super Gödel number of a sequence of wffs that is a PA proof of the wff with Gödel number n . Now de fi ne a wff U ( y ) := ∀ x ¬ Gdl ( x, y ) . Let G = ∃ y ( y = � U ( y ) � ∧ U ( y )) . Now, translating this into English tells us that there exists some natural number y such that y = � U � AND that U ( y ) is true (this is the key!). Think of G as the sentence " G is not provable". – p. 16/1
The Result Suppose G is false. Then, G = ∃ y ( y = � U ( y ) � ∧ U ( y )) is provable which is a contradiction because ∀ x ¬ Gdl ( x, y ) says for all numbers x , none of these numbers is a proof for y , which happens to be the Gödel number for ∀ x ¬ Gdl ( x, y ) itself. Thus, G is true. This statement being true but not provable means that PA must be consistent but incomplete or complete but inconsistent. We want PA to be consistent or we have a trivial inconsistent system. Thus, in a consistent PA, G is true and thus not provable. – p. 17/1
Thank you! Thank you for attending! – p. 18/1
References [1] Franzen, Torkel. Gödel’s Theorem. A K Peters (2005). [2] Gödel, Kurt. On Formally Undecidable Propositions of Principia Mathematica and Related Systems. Dover Publications (1992). [3] Goldstein, Rebecca. Incompleteness: The Proof and Paradox of Kurt Gödel (Great Discoveries). W.W. Norton & Co. (2006). [4] Hofstadter, Douglas. Gödel, Escher, Bach an Eternal Golden Braid; Twentieth Anniversary Edition (1999). [5] Nagel, E., Newman, J. Gödel’s Proof. NYU Press; Revised Edition (2001). [6] B. Russel, A. Whitehead. Principia Mathematica to *56. Cambridge University Press (2008). [7] Smith, Peter. An Introduction to Gödel’s Theorems. Cambridge University Press (2007). [8] Szudzik, Matthew and Weisstein, Eric W. "Parallel Postulate." From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/ParallelPostulate.html. [9] Weisstein, Eric W. "Euclid’s Postulates." From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/EuclidsPostulates.html. – p. 19/1
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