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Ideas of Kurt Gödel, by Text

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

1930 Completeness of Axioms of Logic
p.36 Gödel proved the completeness of first order predicate logic in 1930 [Walicki]
1930 works
p.1 Gödel's Theorems did not refute the claim that all good mathematical questions have answers [Koellner]
p.2 Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable [Koellner]
p.12 For clear questions posed by reason, reason can also find clear answers
p.136 Gödel proved that first-order logic is complete, and second-order logic incomplete [Dummett]
p.182 The real reason for Incompleteness in arithmetic is inability to define truth in a language
p.254 Originally truth was viewed with total suspicion, and only demonstrability was accepted
1931 On Formally Undecidable Propositions
p.3 If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Halbach/Leigh]
p.4 The limitations of axiomatisation were revealed by the incompleteness theorems [Koellner]
p.5 First Incompleteness: arithmetic must always be incomplete [Smith,P]
p.6 Second Incompleteness: nice theories can't prove their own consistency [Smith,P]
p.6 The undecidable sentence can be decided at a 'higher' level in the system
p.104 Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Nagel/Newman]
p.128 Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Shapiro]
p.134 Gödel's Second says that semantic consequence outruns provability [Hanna]
p.157 Prior to Gödel we thought truth in mathematics consisted in provability [Quine]
p.161 First Incompleteness: a decent consistent system is syntactically incomplete [George/Velleman]
p.165 Second Incompleteness: a decent consistent system can't prove its own consistency [George/Velleman]
p.173 There is a sentence which a theory can show is true iff it is unprovable [Smith,P]
p.202 'This system can't prove this statement' makes it unprovable either way [Clegg]
p.212 Basic logic can be done by syntax, with no semantics [Rey]
p.215 There can be no single consistent theory from which all mathematical truths can be derived [George/Velleman]
p.224 Gödel showed that arithmetic is either incomplete or inconsistent [Rey]
p.343 Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P]
p.1215 Gödel show that the incompleteness of set theory was a necessity [Hallett,M]
1944 Russell's Mathematical Logic
n 13 p.455 Impredicative Definitions refer to the totality to which the object itself belongs
p.140-1 p.91 In simple type theory the axiom of Separation is better than Reducibility [Linsky,B]
p.447 p.447 Mathematical Logic is a non-numerical branch of mathematics, and the supreme science
p.448 p.448 A logical system needs a syntactical survey of all possible expressions
p.449 p.449 Some arithmetical problems require assumptions which transcend arithmetic
p.455 p.455 Reference to a totality need not refer to a conjunction of all its elements
p.456 p.456 Mathematical objects are as essential as physical objects are for perception
p.464 p.464 The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers
p.464 p.464 Impredicative definitions are admitted into ordinary mathematics
1964 What is Cantor's Continuum Problem?
p.203 The Continuum Hypothesis is not inconsistent with the axioms of set theory [Clegg]
p.273 If set theory is consistent, we cannot refute or prove the Continuum Hypothesis [Hart,WD]
p.304 Gödel proved the classical relative consistency of the axiom V = L [Putnam]
p.271 p.63 Set-theory paradoxes are no worse than sense deception in physics
p.483 p.35 We perceive the objects of set theory, just as we perceive with our senses
Suppl p.484 Basic mathematics is related to abstract elements of our empirical ideas