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Godel's Theory - Semiotic Version

From Casti (1991)

- Informal version

Arithmetic is not completely formalizable. (p.371) - Formal Logic Version

For every consistent formalization of airthmetic, there exist arithmetic truths that are not provable within that formal system. (p.381)

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

- 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.
- Epimenides Paradox: Replace the notion of 'truth' with that of 'provability', thereby translating the Epimenides Paradox into the assertion 'This statement is unprovable'.
- Godel Sentence: Show that the sentence 'This statement is unprovable' has an arithmetical counterpart, its Godel sentence G, in every conceivable formalization of arithmetic.
- Incompleteness: Prove that the Godel sentence G, must be true if the formal system is consistent.
- 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.
- 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.

Links...

- Implications of Godel's Theory
- Mathematics
- Podnieks, K. (??) "Around the Goedel's theorem" LINK to his on-line book.
- Godel
- Magritte links Hofstadter to Foucault

Created 23/2/98

Modified 3/3/99