1930 | works |
p.1 | 17883 | Gödel's Theorems did not refute the claim that all good mathematical questions have answers |
p.2 | 17885 | Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable |
p.12 | 17892 | For clear questions posed by reason, reason can also find clear answers |
p.136 | 9188 | Gödel proved that first-order logic is complete, and second-order logic incomplete |
p.182 | 10614 | The real reason for Incompleteness in arithmetic is inability to define truth in a language |
p.254 | 10620 | Originally truth was viewed with total suspicion, and only demonstrability was accepted |
1931 | On Formally Undecidable Propositions |
p.4 | 17886 | The limitations of axiomatisation were revealed by the incompleteness theorems |
p.5 | 10072 | First Incompleteness: arithmetic must always be incomplete |
p.6 | 10071 | Second Incompleteness: nice theories can't prove their own consistency |
p.6 | 17888 | The undecidable sentence can be decided at a 'higher' level in the system |
p.104 | 9590 | Arithmetical truth cannot be fully and formally derived from axioms and inference rules |
p.161 | 10118 | First Incompleteness: a decent consistent system is syntactically incomplete |
p.165 | 10122 | Second Incompleteness: a decent consistent system can't prove its own consistency |
p.173 | 10611 | There is a sentence which a theory can show is true iff it is unprovable |
p.202 | 10867 | 'This system can't prove this statement' makes it unprovable either way |
p.215 | 10132 | There can be no single consistent theory from which all mathematical truths can be derived |
1944 | Russell's Mathematical Logic |
n 13 | p.455 | 10041 | Impredicative Definitions refer to the totality to which the object itself belongs |
p.447 | p.447 | 10035 | Mathematical Logic is a non-numerical branch of mathematics, and the supreme science |
p.448 | p.448 | 10038 | A logical system needs a syntactical survey of all possible expressions |
p.449 | p.449 | 10039 | Some arithmetical problems require assumptions which transcend arithmetic |
p.455 | p.455 | 10042 | Reference to a totality need not refer to a conjunction of all its elements |
p.456 | p.456 | 10043 | Mathematical objects are as essential as physical objects are for perception |
p.464 | p.464 | 10045 | Impredicative definitions are admitted into ordinary mathematics |
p.464 | p.464 | 10046 | The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers |
1964 | What is Cantor's Continuum Problem? |
p.271 | p.63 | 18062 | Set-theory paradoxes are no worse than sense deception in physics |
p.483 | p.35 | 8679 | We perceive the objects of set theory, just as we perceive with our senses |
Suppl | p.484 | 10271 | Basic mathematics is related to abstract elements of our empirical ideas |