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

3. Truth / F. Semantic Truth / 1. Tarski's Truth / a. Tarski's truth definition
21752

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

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]

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
21716

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

5. Theory of Logic / A. Overview of Logic / 7. SecondOrder Logic
9188

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

5. Theory of Logic / A. Overview of Logic / 8. Logic of Mathematics
10035

Mathematical Logic is a nonnumerical 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 can't be proved internally, 'reflection principles' can be added to 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

Settheory 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

10868

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

13517

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

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

10072

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

3198

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

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]
