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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic

[discovery that axioms can't prove all truths of arithmetic]

20 ideas
We can add Reflexion Principles to Peano Arithmetic, which assert its consistency or soundness [Halbach on Peano]
Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey]
First Incompleteness: arithmetic must always be incomplete [Gödel, by Smith,P]
Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Gödel, by Nagel/Newman]
Gödel's Second says that semantic consequence outruns provability [Gödel, by Hanna]
First Incompleteness: a decent consistent system is syntactically incomplete [Gödel, by George/Velleman]
Second Incompleteness: a decent consistent system can't prove its own consistency [Gödel, by George/Velleman]
There is a sentence which a theory can show is true iff it is unprovable [Gödel, by Smith,P]
'This system can't prove this statement' makes it unprovable either way [Gödel, by Clegg]
Some arithmetical problems require assumptions which transcend arithmetic [Gödel]
Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable [Gödel, by Koellner]
The real reason for Incompleteness in arithmetic is inability to define truth in a language [Gödel]
Gentzen proved the consistency of arithmetic from assumptions beyond arithmetic [Gentzen, by Musgrave]
Intuitionists find the Incompleteness Theorem unsurprising, since proof is intuitive, not formal [Dummett]
Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P]
Multiplication only generates incompleteness if combined with addition and successor [Smith,P]
It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry]
The incompletability of formal arithmetic reveals that logic also cannot be completely characterized [Hale/Wright]
The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman]
Arithmetical undecidability is always settled at the next stage up [Koellner]