- This must mean that it is impossible to characterize arithmetic fully in terms of first-order logic. In which case we can only characterize the features of abstract reality in general if we employ an incomplete system. We're doomed.
COMMENT: Materialism is doomed yes, because the universe is held together by "calling those things that be not as though they were" on a continual basis.
Truth can't be defined Edit
- ....The real reason for Incompleteness in arithmetic is inability to define truth in a language...'
21 ideas Edit
The 21 ideas from Kurt Gödel 8679 We perceive the objects of set theory, just as we perceive with our senses [Gödel] 10271 Basic mathematics is related to abstract elements of our empirical ideas [Gödel] 10611 There is a sentence which a theory can show is true iff it is unprovable [Smith,P on Gödel] 10122 Second Incompleteness: a decent consistent system can't prove its own consistency [George/Velleman on Gödel] 10867 'This system can't prove this statement' makes it unprovable either way [Clegg on Gödel] 10072 First Incompleteness: arithmetic must always be incomplete [Smith,P on Gödel] 9590 Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Nagel/Newman on Gödel] 10118 First Incompleteness: a decent consistent system is syntactically incomplete [George/Velleman on Gödel] 10132 There can be no single consistent theory from which all mathematical truths can be derived [George/Velleman on Gödel] 10071 Second Incompleteness: nice theories can't prove their own consistency [Smith,P on Gödel] 10041 Impredicative Definitions refer to the totality to which the object itself belongs [Gödel] 10035 Mathematical Logic is a non-numerical branch of mathematics, and the supreme science [Gödel] 10038 A logical system needs a syntactical survey of all possible expressions [Gödel] 10039 Some arithmetical problems require assumptions which transcend arithmetic [Gödel] 10042 Reference to a totality need not refer to a conjunction of all its elements [Gödel] 10043 Mathematical objects are as essential as physical objects are for perception [Gödel] 10045 Impredicative definitions are admitted into ordinary mathematics [Gödel] 10046 The generalized Continuum Hypothesis assserts a gap in cardinal numbers [Gödel] 9188 Gödel proved that first-order logic is complete, and second-order logic incomplete [Dummett on Gödel] 10620 Originally truth was viewed with total suspicion, and only demonstrability was accepted [Gödel] 10614 The real reason for Incompleteness in arithmetic is inability to define truth in a language [Gödel] email your comments