Combining Texts

All the ideas for 'fragments/reports', 'Miscellaneous Observations' and 'On Formally Undecidable Propositions'

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27 ideas

3. Truth / A. Truth Problems / 3. Value of Truth
19574 If man sacrifices truth he sacrifices himself, by acting against his own convictions [Novalis]
3. Truth / E. Pragmatic Truth / 1. Pragmatic Truth
19571 Delusion and truth differ in their life functions [Novalis]
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 [Gödel, by Quine]
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 [Gödel, by Hallett,M]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
17886 The limitations of axiomatisation were revealed by the incompleteness theorems [Gödel, by Koellner]
5. Theory of Logic / K. Features of Logics / 2. Consistency
10071 Second Incompleteness: nice theories can't prove their own consistency [Gödel, by 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 [Gödel, by Halbach/Leigh]
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
10621 Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P on Gödel]
17888 The undecidable sentence can be decided at a 'higher' level in the system [Gödel]
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
10132 There can be no single consistent theory from which all mathematical truths can be derived [Gödel, by George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
10072 First Incompleteness: arithmetic must always be incomplete [Gödel, by Smith,P]
9590 Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Gödel, by Nagel/Newman]
3198 Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey]
11069 Gödel's Second says that semantic consequence outruns provability [Gödel, by Hanna]
10118 First Incompleteness: a decent consistent system is syntactically incomplete [Gödel, by George/Velleman]
10122 Second Incompleteness: a decent consistent system can't prove its own consistency [Gödel, by George/Velleman]
10611 There is a sentence which a theory can show is true iff it is unprovable [Gödel, by Smith,P]
10867 'This system can't prove this statement' makes it unprovable either way [Gödel, by Clegg]
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 [Gödel, by Shapiro]
9. Objects / D. Essence of Objects / 3. Individual Essences
19575 Refinement of senses increasingly distinguishes individuals [Novalis]
12. Knowledge Sources / C. Rationalism / 1. Rationalism
19572 Experiences tests reason, and reason tests experience [Novalis]
17. Mind and Body / C. Functionalism / 2. Machine Functionalism
3192 Basic logic can be done by syntax, with no semantics [Gödel, by Rey]
17. Mind and Body / E. Mind as Physical / 1. Physical Mind
19573 The seat of the soul is where our inner and outer worlds interpenetrate [Novalis]
18. Thought / E. Abstraction / 2. Abstracta by Selection
19577 Everything is a chaotic unity, then we abstract, then we reunify the world into a free alliance [Novalis]
21. Aesthetics / A. Aesthetic Experience / 5. Natural Beauty
19578 Only self-illuminated perfect individuals are beautiful [Novalis]
21. Aesthetics / C. Artistic Issues / 7. Art and Morality
468 Musical performance can reveal a range of virtues [Damon of Ath.]
29. Religion / D. Religious Issues / 1. Religious Commitment / a. Religious Belief
19576 Religion needs an intermediary, because none of us can connect directly to a godhead [Novalis]