### Ideas of Kurt Gödel, by Theme

#### [Austrian, 1906 - 1978, Born in Brno, Austria. Ended up at Institute of Advanced Studies at Princeton, with Einstein.]

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###### 2. Reason / A. Nature of Reason / 1. On Reason
 17892 For clear questions posed by reason, reason can also find clear answers
###### 2. Reason / D. Definition / 8. Impredicative Definition
 10041 Impredicative Definitions refer to the totality to which the object itself belongs
###### 4. Formal Logic / C. Predicate Calculus PC / 3. Completeness of PC
 17751 Gödel proved the completeness of first order predicate logic in 1930 [Walicki]
###### 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
 17835 Gödel show that the incompleteness of set theory was a necessity [Hallett,M]
 8679 We perceive the objects of set theory, just as we perceive with our senses
###### 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / o. Axiom of Constructibility V = L
 9942 Gödel proved the classical relative consistency of the axiom V = L [Putnam]
###### 5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
 9188 Gödel proved that first-order logic is complete, and second-order logic incomplete [Dummett]
###### 5. Theory of Logic / A. Overview of Logic / 8. Logic of Mathematics
 10035 Mathematical Logic is a non-numerical branch of mathematics, and the supreme science
###### 5. Theory of Logic / G. Quantification / 2. Domain of Quantification
 10042 Reference to a totality need not refer to a conjunction of all its elements
###### 5. Theory of Logic / I. Semantics of Logic / 2. Formal Truth
 10620 Originally truth was viewed with total suspicion, and only demonstrability was accepted
###### 5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
 17886 The limitations of axiomatisation were revealed by the incompleteness theorems [Koellner]
###### 5. Theory of Logic / K. Features of Logics / 2. Consistency
 10071 Second Incompleteness: nice theories can't prove their own consistency [Smith,P]
###### 5. Theory of Logic / K. Features of Logics / 3. Soundness
 19123 If soundness cannot be proved internally, 'reflection principles' be added which assert soundness [Halbach/Leigh]
###### 5. Theory of Logic / K. Features of Logics / 5. Incompleteness
 17883 Gödel's Theorems did not refute the claim that all good mathematical questions have answers [Koellner]
 17888 The undecidable sentence can be decided at a 'higher' level in the system
 10621 Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P]
###### 5. Theory of Logic / K. Features of Logics / 8. Enumerability
 10038 A logical system needs a syntactical survey of all possible expressions
###### 5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / a. Set theory paradoxes
 18062 Set-theory paradoxes are no worse than sense deception in physics
###### 6. Mathematics / A. Nature of Mathematics / 1. Mathematics
 10132 There can be no single consistent theory from which all mathematical truths can be derived [George/Velleman]
###### 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
 10046 The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers
 13517 If set theory is consistent, we cannot refute or prove the Continuum Hypothesis [Hart,WD]
 10868 The Continuum Hypothesis is not inconsistent with the axioms of set theory [Clegg]
###### 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
 17885 Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable [Koellner]
 10614 The real reason for Incompleteness in arithmetic is inability to define truth in a language
 3198 Gödel showed that arithmetic is either incomplete or inconsistent [Rey]
 10072 First Incompleteness: arithmetic must always be incomplete [Smith,P]
 9590 Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Nagel/Newman]
 11069 Gödel's Second says that semantic consequence outruns provability [Hanna]
 10118 First Incompleteness: a decent consistent system is syntactically incomplete [George/Velleman]
 10122 Second Incompleteness: a decent consistent system can't prove its own consistency [George/Velleman]
 10611 There is a sentence which a theory can show is true iff it is unprovable [Smith,P]
 10867 'This system can't prove this statement' makes it unprovable either way [Clegg]
 10039 Some arithmetical problems require assumptions which transcend arithmetic
###### 6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
 10043 Mathematical objects are as essential as physical objects are for perception
###### 6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
 10271 Basic mathematics is related to abstract elements of our empirical ideas
###### 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
 8747 Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Shapiro]
 10045 Impredicative definitions are admitted into ordinary mathematics
###### 17. Mind and Body / C. Functionalism / 2. Machine Functionalism
 3192 Basic logic can be done by syntax, with no semantics [Rey]