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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

17751

Gödel proved the completeness of first order predicate logic in 1930 [Walicki]


p.1

17883

Gödel's Theorems did not refute the claim that all good mathematical questions have answers [Koellner]


p.2

17885

Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable [Koellner]


p.12

17892

For clear questions posed by reason, reason can also find clear answers


p.136

9188

Gödel proved that firstorder logic is complete, and secondorder logic incomplete [Dummett]


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.3

19123

If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Halbach/Leigh]


p.4

17886

The limitations of axiomatisation were revealed by the incompleteness theorems [Koellner]


p.5

10072

First Incompleteness: arithmetic must always be incomplete [Smith,P]


p.6

10071

Second Incompleteness: nice theories can't prove their own consistency [Smith,P]


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 [Nagel/Newman]


p.128

8747

Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Shapiro]


p.134

11069

Gödel's Second says that semantic consequence outruns provability [Hanna]


p.157

21752

Prior to Gödel we thought truth in mathematics consisted in provability [Quine]


p.161

10118

First Incompleteness: a decent consistent system is syntactically incomplete [George/Velleman]


p.165

10122

Second Incompleteness: a decent consistent system can't prove its own consistency [George/Velleman]


p.173

10611

There is a sentence which a theory can show is true iff it is unprovable [Smith,P]


p.202

10867

'This system can't prove this statement' makes it unprovable either way [Clegg]


p.212

3192

Basic logic can be done by syntax, with no semantics [Rey]


p.215

10132

There can be no single consistent theory from which all mathematical truths can be derived [George/Velleman]


p.224

3198

Gödel showed that arithmetic is either incomplete or inconsistent [Rey]


p.343

10621

Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P]


p.1215

17835

Gödel show that the incompleteness of set theory was a necessity [Hallett,M]

1944

Russell's Mathematical Logic

n 13

p.455

10041

Impredicative Definitions refer to the totality to which the object itself belongs

p.1401

p.91

21716

In simple type theory the axiom of Separation is better than Reducibility [Linsky,B]

p.447

p.447

10035

Mathematical Logic is a nonnumerical 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

10046

The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers

p.464

p.464

10045

Impredicative definitions are admitted into ordinary mathematics

1964

What is Cantor's Continuum Problem?


p.203

10868

The Continuum Hypothesis is not inconsistent with the axioms of set theory [Clegg]


p.273

13517

If set theory is consistent, we cannot refute or prove the Continuum Hypothesis [Hart,WD]


p.304

9942

Gödel proved the classical relative consistency of the axiom V = L [Putnam]

p.271

p.63

18062

Settheory 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
