Ideas from 'On Formally Undecidable Propositions' by Kurt Gödel [1931], by Theme Structure

[found in 'From Frege to Gödel 1879-1931' (ed/tr Heijenoort,Jean van) [Harvard 1967,0-674-32449-8]].

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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
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]
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 cannot be proved internally, 'reflection principles' be added which assert soundness [Halbach/Leigh]
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P]
The undecidable sentence can be decided at a 'higher' level in the system
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 / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
First Incompleteness: arithmetic must always be incomplete [Smith,P]
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]
Gödel showed that arithmetic is either incomplete or inconsistent [Rey]
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]
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]
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]
17. Mind and Body / C. Functionalism / 2. Machine Functionalism
Basic logic can be done by syntax, with no semantics [Rey]