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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.
The main steps in Godel's proof. Casti (1991,
- 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
- 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.