Combining Texts

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

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25 ideas

2. Reason / A. Nature of Reason / 1. On Reason
6950 You can be rational with undetected or minor inconsistencies [Harman]
2. Reason / A. Nature of Reason / 6. Coherence
6954 A coherent conceptual scheme contains best explanations of most of your beliefs [Harman]
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]
14. Science / C. Induction / 1. Induction
6955 Enumerative induction is inference to the best explanation [Harman]
14. Science / C. Induction / 3. Limits of Induction
6952 Induction is 'defeasible', since additional information can invalidate it [Harman]
14. Science / C. Induction / 4. Reason in Induction
6953 All reasoning is inductive, and deduction only concerns implication [Harman]
17. Mind and Body / C. Functionalism / 2. Machine Functionalism
3192 Basic logic can be done by syntax, with no semantics [Gödel, by Rey]
18. Thought / A. Modes of Thought / 5. Rationality / a. Rationality
6951 Ordinary rationality is conservative, starting from where your beliefs currently are [Harman]
20. Action / B. Preliminaries of Action / 2. Willed Action / d. Weakness of will
22511 Some reasonings are stronger than we are [Philolaus]