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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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
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 / B. Foundations for Mathematics / 3. Axioms for Number / g. Incompleteness of Arithmetic
There is a sentence which a theory can show is true iff it is unprovable
First Incompleteness: arithmetic must always be incomplete
Arithmetical truth cannot be fully and formally derived from axioms and inference rules
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