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
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
We perceive the objects of set theory, just as we perceive with our senses
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
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
5. Theory of Logic / K. Features of Logics / 2. Consistency
Second Incompleteness: nice theories can't prove their own consistency
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
The undecidable sentence can be decided at a 'higher' level in the system
Gödel's Theorems did not refute the claim that all good mathematical questions have answers
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
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / g. Continuum Hypothesis
The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / g. Incompleteness of Arithmetic
Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable
The real reason for Incompleteness in arithmetic is inability to define truth in a language
First Incompleteness: arithmetic must always be incomplete
Arithmetical truth cannot be fully and formally derived from axioms and inference rules
There is a sentence which a theory can show is true iff it is unprovable
First Incompleteness: a decent consistent system is syntactically incomplete
Second Incompleteness: a decent consistent system can't prove its own consistency
'This system can't prove this statement' makes it unprovable either way
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
Impredicative definitions are admitted into ordinary mathematics