Godel's Theory

(4/7/99 - see Trees and graphs for a different approach to this problem. Most of the attributions made here are tentative)

# 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.
* Signification: A sign consists of a signifier and a signified

* Barthes (1957:1972) "Mythologies". A myth is a second level of signification. A mythological sign also consists of a signifier and a signified, but here the signifier is also a language sign (p.115).

Casti's use of 'mirror-image' is a bit vague. Better terminology is available from linguistics - namely the reflective, intentional, and constructive models of language.

- # Epimenides Paradox: Replace the notion of 'truth' with that of
'provability', thereby translating the Epimenides Paradox into the
assertion 'This statement is unprovable'.
Formal, axiomatic systems have axioms and theorems. Axioms are theorems which are assumed to be true for the particular formal systems. Theorems are then true or false depending on the formal system used.

* Symbolic systems cannot be assumed to give absolute meaning, meaning is contextual to the particular formal structure.

- # 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 concrete fixed model (theory) can solve all problems arising in science (or even in mathematics itself). An excellent confirmation of this thesis was given in the famous incompleteness theorem of K. Goedel." Podnieks, K. (??)

- # 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...

- Mathematics
- Podnieks, K. (??) "Around the Goedel's theorem" LINK to his on-line book.

Created 27/2/98

Modified 5/4/99