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Single Idea 10128

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic ]

Full Idea

The idea behind the proofs of the Incompleteness Theorems is to use the language of Peano Arithmetic to talk about the formal system of Peano Arithmetic itself.

Gist of Idea

The Incompleteness proofs use arithmetic to talk about formal arithmetic

Source

A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7)

Book Ref

George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.182

A Reaction

The mechanism used is to assign a Gödel Number to every possible formula, so that all reasonings become instances of arithmetic.

The 21 ideas with the same theme [discovery that axioms can't prove all truths of arithmetic]:

15653 We can add Reflexion Principles to Peano Arithmetic, which assert its consistency or soundness [Halbach on Peano]
3198 Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey]
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]
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]
10039 Some arithmetical problems require assumptions which transcend arithmetic [Gödel]
17885 Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable [Gödel, by Koellner]
10614 The real reason for Incompleteness in arithmetic is inability to define truth in a language [Gödel]
10067 Gentzen proved the consistency of arithmetic from assumptions beyond arithmetic [Gentzen, by Musgrave]
10554 Intuitionists find the Incompleteness Theorem unsurprising, since proof is intuitive, not formal [Dummett]
10604 Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P]
10848 Multiplication only generates incompleteness if combined with addition and successor [Smith,P]
17793 It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry]
10624 The incompletability of formal arithmetic reveals that logic also cannot be completely characterized [Hale/Wright]
10128 The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman]
17891 Arithmetical undecidability is always settled at the next stage up [Koellner]
23446 You can't prove consistency using a weaker theory, but you can use a consistent theory [Linnebo]