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Single Idea 10072

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

Full Idea

First Incompleteness Theorem: any properly axiomatised and consistent theory of basic arithmetic must remain incomplete, whatever our efforts to complete it by throwing further axioms into the mix.

Gist of Idea

First Incompleteness: arithmetic must always be incomplete

Source

report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 1.2

Book Ref

Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.5

A Reaction

This is because it is always possible to formulate a well-formed sentence which is not provable within the theory.

The 18 ideas from 'On Formally Undecidable Propositions'

21752 Prior to Gödel we thought truth in mathematics consisted in provability [Gödel, by Quine]
17835 Gödel show that the incompleteness of set theory was a necessity [Gödel, by Hallett,M]
10071 Second Incompleteness: nice theories can't prove their own consistency [Gödel, by Smith,P]
17886 The limitations of axiomatisation were revealed by the incompleteness theorems [Gödel, by Koellner]
19123 If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Gödel, by Halbach/Leigh]
10621 Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P on Gödel]
17888 The undecidable sentence can be decided at a 'higher' level in the system [Gödel]
10132 There can be no single consistent theory from which all mathematical truths can be derived [Gödel, by George/Velleman]
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]
8747 Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro]
3192 Basic logic can be done by syntax, with no semantics [Gödel, by Rey]