- # 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
* 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
- # 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.