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Single Idea 21752
[filed under theme 3. Truth / F. Semantic Truth / 1. Tarski's Truth / a. Tarski's truth definition
]
Full Idea
Gödel's proof wrought an abrupt turn in the philosophy of mathematics. We had supposed that truth, in mathematics, consisted in provability.
Gist of Idea
Prior to Gödel we thought truth in mathematics consisted in provability
Source
report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Willard Quine - Forward to Gödel's Unpublished
Book Ref
Quine,Willard: 'W.V. Quine: Quine in Dialogue' [Harvard 2008], p.157
A Reaction
This explains the crisis in the early 1930s, which Tarski's theory appeared to solve.
The
18 ideas
from 'On Formally Undecidable Propositions'
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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