Ideas from 'On Formally Undecidable Propositions' by Kurt Gödel [1931], by Theme Structure
[found in 'From Frege to Gödel 18791931' (ed/tr Heijenoort,Jean van) [Harvard 1967,0674324498]].
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5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
17886

The limitations of axiomatisation were revealed by the incompleteness theorems

5. Theory of Logic / K. Features of Logics / 2. Consistency
10071

Second Incompleteness: nice theories can't prove their own consistency

5. Theory of Logic / K. Features of Logics / 5. Incompleteness
17888

The undecidable sentence can be decided at a 'higher' level in the system

6. Mathematics / A. Nature of Mathematics / 1. Mathematics
10132

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
10072

First Incompleteness: arithmetic must always be incomplete

9590

Arithmetical truth cannot be fully and formally derived from axioms and inference rules

10118

First Incompleteness: a decent consistent system is syntactically incomplete

10122

Second Incompleteness: a decent consistent system can't prove its own consistency

10611

There is a sentence which a theory can show is true iff it is unprovable

10867

'This system can't prove this statement' makes it unprovable either way
