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Single Idea 9590
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
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Full Idea
The vast continent of arithmetical truth cannot be brought into systematic order by laying down a fixed set of axioms and rules of inference from which every true mathematical statement can be formally derived. For some this was a shocking revelation.
Gist of Idea
Arithmetical truth cannot be fully and formally derived from axioms and inference rules
Source
report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by E Nagel / JR Newman - Gödel's Proof VII.C
Book Ref
Nagel,E/Newman,J.R.: 'Gödel's Proof' [NYU 2001], p.104
A Reaction
Good news for philosophy, I'd say. The truth cannot be worked out by mechanical procedures, so it needs the subtle and intuitive intelligence of your proper philosopher (Parmenides is the role model) to actually understand reality.
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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