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
For clear questions posed by reason, reason can also find clear answers
2. Reason / D. Definition / 8. Impredicative Definition
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
Prior to Gödel we thought truth in mathematics consisted in provability [Quine]
4. Formal Logic / C. Predicate Calculus PC / 3. Completeness of PC
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
Gödel show that the incompleteness of set theory was a necessity [Hallett,M]
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
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
In simple type theory the axiom of Separation is better than Reducibility [Linsky,B]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Gödel proved that first-order logic is complete, and second-order logic incomplete [Dummett]
5. Theory of Logic / A. Overview of Logic / 8. Logic of Mathematics
Mathematical Logic is a non-numerical branch of mathematics, and the supreme science
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
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
Originally truth was viewed with total suspicion, and only demonstrability was accepted
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
The limitations of axiomatisation were revealed by the incompleteness theorems [Koellner]
5. Theory of Logic / K. Features of Logics / 2. Consistency
Second Incompleteness: nice theories can't prove their own consistency [Smith,P]
5. Theory of Logic / K. Features of Logics / 3. Soundness
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
Gödel's Theorems did not refute the claim that all good mathematical questions have answers [Koellner]
The undecidable sentence can be decided at a 'higher' level in the system
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
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
Set-theory paradoxes are no worse than sense deception in physics
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
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
The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers
The Continuum Hypothesis is not inconsistent with the axioms of set theory [Clegg]
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
Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable [Koellner]
The real reason for Incompleteness in arithmetic is inability to define truth in a language
First Incompleteness: arithmetic must always be incomplete [Smith,P]
Gödel showed that arithmetic is either incomplete or inconsistent [Rey]
Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Nagel/Newman]
Gödel's Second says that semantic consequence outruns provability [Hanna]
First Incompleteness: a decent consistent system is syntactically incomplete [George/Velleman]
Second Incompleteness: a decent consistent system can't prove its own consistency [George/Velleman]
There is a sentence which a theory can show is true iff it is unprovable [Smith,P]
'This system can't prove this statement' makes it unprovable either way [Clegg]
Some arithmetical problems require assumptions which transcend arithmetic
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
Mathematical objects are as essential as physical objects are for perception
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
Basic mathematics is related to abstract elements of our empirical ideas
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Shapiro]
Impredicative definitions are admitted into ordinary mathematics
17. Mind and Body / C. Functionalism / 2. Machine Functionalism
Basic logic can be done by syntax, with no semantics [Rey]