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Godel's Theory
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.
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Created 23/2/98
Modified 3/3/99