#concept
godel’s 2 incompleteness theorems are concerned with >> the limits of provability in formal axiomatic theories.
Any system that can talk about itself and its own consistency is necessarily inconsistent.
First Incompleteness Theorem >> “Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.” [1]
Second Incompleteness Theorem >> “Assume F is a consistent formalized system which contains elementary arithmetic. Then F does not prove the consistency of F” [2]
True or False: Godel’s first theorem shows there are truths that can’t be proved. Explain ? False. Incompleteness theorem doesn’t deal with provability in an absolute sense, but rather the derivability in a specific formal system. Statement X may be unprovable in formal system A but provable in formal system B.
According to Gödel’s incompleteness theorems, the statement that Peano Arithmetic (PA) is consistent, in its guise as a number-theoretic statement (given the technique of godel numbering) ? cannot be derived in PA itself. godel numbering is used to prove these theorems.
References
Notes
1st theorem: not complete
- any sentence can be represented w a godel number
- any proof can be represented w a godel number
- the proof referring to itself cannot be represented as a godel number
- g: there is no proof with the statement godel g
2nd: can’t prove consistency
my hot takes
but…you kinda got to accept formal system of peano arithmetic, to accept godel numbering… i just think a formal system itself is flawed…logical system…