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Single Idea 10118

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

Full Idea

First Incompleteness Theorem: If S is a sufficiently powerful formal system, then if S is consistent then S is syntactically incomplete.

Gist of Idea

First Incompleteness: a decent consistent system is syntactically incomplete

Source

report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6

Book Ref

George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.161

A Reaction

Gödel found a single sentence, effectively saying 'I am unprovable in S', which is neither provable nor refutable in S.

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]