more from this thinker     |     more from this text

Single Idea 10611

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

Full Idea

The original Gödel construction gives us a sentence that a theory shows is true if and only if it satisfies the condition of being unprovable-in-that-theory.

Gist of Idea

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

Source

report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 20.5

Book Ref

Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.173

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]
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]
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]
3198 Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey]
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]
email your comments