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Godel's Theory


Godel's Theory - Semiotic Version

From Casti (1991)
The main steps in Godel's proof. Casti (1991, p.382)

  1. Godel Numbering: Development of a coding scheme to translate everylogical formula and proof sequence in Principia Mathematica into a 'mirror-image' statement about the natural numbers.
  2. Epimenides Paradox: Replace the notion of 'truth' with that of 'provability', thereby translating the Epimenides Paradox into the assertion 'This statement is unprovable'.
  3. Godel Sentence: Show that the sentence 'This statement is unprovable' has an arithmetical counterpart, its Godel sentence G, in every conceivable formalization of arithmetic.
  4. Incompleteness: Prove that the Godel sentence G, must be true if the formal system is consistent.
  5. No Escape Clause: Prove that even if additional axioms are added to form a new system in which G is provable, the new system with the additional axioms will have its own unprovable Godel sentence.
  6. Consistency: Construct an arithmetical statement asserting that 'arithmetic is consistent'. Prove that this arithmetical statement is not provable, thus showing that arithmetic as a formal system is too weak to prove its own consistency.

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Created 23/2/98
Modified 3/3/99