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Single Idea 10071
[filed under theme 5. Theory of Logic / K. Features of Logics / 2. Consistency
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Full Idea
Second Incompleteness Theorem: roughly, nice theories that include enough basic arithmetic can't prove their own consistency.
Gist of Idea
Second Incompleteness: nice theories can't prove their own consistency
Source
report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 1.5
Book Ref
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.6
A Reaction
On the face of it, this sounds less surprising than the First Theorem. Philosophers have often noticed that it seems unlikely that you could use reason to prove reason, as when Descartes just relies on 'clear and distinct ideas'.
The
18 ideas
from 'On Formally Undecidable Propositions'
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21752
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Prior to Gödel we thought truth in mathematics consisted in provability
[Gödel, by Quine]
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17835
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Gödel show that the incompleteness of set theory was a necessity
[Gödel, by Hallett,M]
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10071
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Second Incompleteness: nice theories can't prove their own consistency
[Gödel, by Smith,P]
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17886
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The limitations of axiomatisation were revealed by the incompleteness theorems
[Gödel, by Koellner]
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19123
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If soundness can't be proved internally, 'reflection principles' can be added to assert soundness
[Gödel, by Halbach/Leigh]
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10621
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Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme
[Smith,P on Gödel]
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17888
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The undecidable sentence can be decided at a 'higher' level in the system
[Gödel]
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10132
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There can be no single consistent theory from which all mathematical truths can be derived
[Gödel, by George/Velleman]
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3198
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Gödel showed that arithmetic is either incomplete or inconsistent
[Gödel, by Rey]
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10072
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First Incompleteness: arithmetic must always be incomplete
[Gödel, by Smith,P]
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9590
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Arithmetical truth cannot be fully and formally derived from axioms and inference rules
[Gödel, by Nagel/Newman]
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11069
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Gödel's Second says that semantic consequence outruns provability
[Gödel, by Hanna]
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10118
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First Incompleteness: a decent consistent system is syntactically incomplete
[Gödel, by George/Velleman]
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10122
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Second Incompleteness: a decent consistent system can't prove its own consistency
[Gödel, by George/Velleman]
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10611
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There is a sentence which a theory can show is true iff it is unprovable
[Gödel, by Smith,P]
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10867
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'This system can't prove this statement' makes it unprovable either way
[Gödel, by Clegg]
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8747
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Realists are happy with impredicative definitions, which describe entities in terms of other existing entities
[Gödel, by Shapiro]
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3192
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Basic logic can be done by syntax, with no semantics
[Gödel, by Rey]
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