Philosophyideas.com

=== Godel ===
http://www.philosophyideas.com/css/search/idea_detail.asp?find=idea&visit=2&ThemeNumber=1401&area=Formal%20Logic&area_no=3&ID=9188&return=yes&theme_alpha=yes&gistsfor=idea&source=theme
: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 ===
http://www.philosophyideas.com/css/search/idea_detail.asp?find=idea&visit=2&ThemeNumber=1401&area=Formal%20Logic&area_no=3&ID=10614&return=yes&theme_alpha=yes&gistsfor=idea&source=theme
:....The real reason for Incompleteness in arithmetic is inability to define truth in a language...'
=== 21 ideas ===
http://www.philosophyideas.com/css/search/idea_detail.asp?find=theme&themenumber=1401&visit=1&return=yes&ID=10611&gistsfor=philosopher&source=theme&area=Formal%20Logic&area_no=3
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
Category:Godel
Category:Philosophy