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Single Idea 10122
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
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Full Idea
Second Incompleteness Theorem: If S is a sufficiently powerful formal system, then if S is consistent then S cannot prove its own consistency
Gist of Idea
Second Incompleteness: a decent consistent system can't prove its own consistency
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.165
A Reaction
This seems much less surprising than the First Theorem (though it derives from it). It was always kind of obvious that you couldn't use reason to prove that reason works (see, for example, the Cartesian Circle).
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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17886
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The limitations of axiomatisation were revealed by the incompleteness theorems
[Gödel, by Koellner]
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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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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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17888
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The undecidable sentence can be decided at a 'higher' level in the system
[Gödel]
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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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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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10072
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First Incompleteness: arithmetic must always be incomplete
[Gödel, by Smith,P]
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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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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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