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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.]
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1930
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Completeness of Axioms of Logic
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p.36
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17751
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Gödel proved the completeness of first order predicate logic in 1930 [Walicki]
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p.1
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17883
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Gödel's Theorems did not refute the claim that all good mathematical questions have answers [Koellner]
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p.2
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17885
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Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable [Koellner]
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p.12
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17892
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For clear questions posed by reason, reason can also find clear answers
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p.136
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9188
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Gödel proved that first-order logic is complete, and second-order logic incomplete [Dummett]
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p.182
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10614
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The real reason for Incompleteness in arithmetic is inability to define truth in a language
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p.254
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10620
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Originally truth was viewed with total suspicion, and only demonstrability was accepted
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1931
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On Formally Undecidable Propositions
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p.3
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19123
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If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Halbach/Leigh]
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p.4
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17886
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The limitations of axiomatisation were revealed by the incompleteness theorems [Koellner]
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p.5
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10072
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First Incompleteness: arithmetic must always be incomplete [Smith,P]
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p.6
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10071
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Second Incompleteness: nice theories can't prove their own consistency [Smith,P]
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p.6
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17888
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The undecidable sentence can be decided at a 'higher' level in the system
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p.104
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9590
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Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Nagel/Newman]
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p.128
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8747
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Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Shapiro]
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p.134
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11069
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Gödel's Second says that semantic consequence outruns provability [Hanna]
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p.157
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21752
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Prior to Gödel we thought truth in mathematics consisted in provability [Quine]
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p.161
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10118
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First Incompleteness: a decent consistent system is syntactically incomplete [George/Velleman]
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p.165
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10122
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Second Incompleteness: a decent consistent system can't prove its own consistency [George/Velleman]
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p.173
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10611
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There is a sentence which a theory can show is true iff it is unprovable [Smith,P]
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p.202
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10867
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'This system can't prove this statement' makes it unprovable either way [Clegg]
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p.212
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3192
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Basic logic can be done by syntax, with no semantics [Rey]
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p.215
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10132
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There can be no single consistent theory from which all mathematical truths can be derived [George/Velleman]
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p.224
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3198
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Gödel showed that arithmetic is either incomplete or inconsistent [Rey]
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p.343
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10621
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Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P]
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p.1215
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17835
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Gödel show that the incompleteness of set theory was a necessity [Hallett,M]
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1944
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Russell's Mathematical Logic
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n 13
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p.455
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10041
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Impredicative Definitions refer to the totality to which the object itself belongs
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p.140-1
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p.91
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21716
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In simple type theory the axiom of Separation is better than Reducibility [Linsky,B]
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p.447
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p.447
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10035
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Mathematical Logic is a non-numerical branch of mathematics, and the supreme science
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p.448
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p.448
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10038
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A logical system needs a syntactical survey of all possible expressions
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p.449
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p.449
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10039
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Some arithmetical problems require assumptions which transcend arithmetic
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p.455
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p.455
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10042
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Reference to a totality need not refer to a conjunction of all its elements
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p.456
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p.456
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10043
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Mathematical objects are as essential as physical objects are for perception
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p.464
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p.464
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10046
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The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers
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p.464
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p.464
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10045
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Impredicative definitions are admitted into ordinary mathematics
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1964
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What is Cantor's Continuum Problem?
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p.203
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10868
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The Continuum Hypothesis is not inconsistent with the axioms of set theory [Clegg]
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p.273
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13517
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If set theory is consistent, we cannot refute or prove the Continuum Hypothesis [Hart,WD]
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p.304
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9942
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Gödel proved the classical relative consistency of the axiom V = L [Putnam]
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p.271
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p.63
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18062
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Set-theory paradoxes are no worse than sense deception in physics
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p.483
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p.35
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8679
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We perceive the objects of set theory, just as we perceive with our senses
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Suppl
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p.484
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10271
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Basic mathematics is related to abstract elements of our empirical ideas
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