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Single Idea 17793
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
]
Full Idea
If we did not know that the second-order axioms characterise the natural numbers up to isomorphism, we should have no reason to suppose, a priori, that first-order Peano Arithmetic should be complete.
Gist of Idea
It is only 2nd-order isomorphism which suggested first-order PA completeness
Source
John Mayberry (What Required for Foundation for Maths? [1994], p.412-1)
Book Ref
'Philosophy of Mathematics: anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.412
The
21 ideas
with the same theme
[discovery that axioms can't prove all truths of arithmetic]:
15653
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We can add Reflexion Principles to Peano Arithmetic, which assert its consistency or soundness
[Halbach on Peano]
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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]
|
11069
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Gödel's Second says that semantic consequence outruns provability
[Gödel, by Hanna]
|
10118
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First Incompleteness: a decent consistent system is syntactically incomplete
[Gödel, by George/Velleman]
|
10122
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Second Incompleteness: a decent consistent system can't prove its own consistency
[Gödel, by George/Velleman]
|
10611
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There is a sentence which a theory can show is true iff it is unprovable
[Gödel, by Smith,P]
|
10867
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'This system can't prove this statement' makes it unprovable either way
[Gödel, by Clegg]
|
10039
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Some arithmetical problems require assumptions which transcend arithmetic
[Gödel]
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17885
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Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable
[Gödel, by Koellner]
|
10614
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The real reason for Incompleteness in arithmetic is inability to define truth in a language
[Gödel]
|
10067
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Gentzen proved the consistency of arithmetic from assumptions beyond arithmetic
[Gentzen, by Musgrave]
|
10554
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Intuitionists find the Incompleteness Theorem unsurprising, since proof is intuitive, not formal
[Dummett]
|
10604
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Incompleteness results in arithmetic from combining addition and successor with multiplication
[Smith,P]
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10848
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Multiplication only generates incompleteness if combined with addition and successor
[Smith,P]
|
17793
|
It is only 2nd-order isomorphism which suggested first-order PA completeness
[Mayberry]
|
10624
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The incompletability of formal arithmetic reveals that logic also cannot be completely characterized
[Hale/Wright]
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10128
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The Incompleteness proofs use arithmetic to talk about formal arithmetic
[George/Velleman]
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17891
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Arithmetical undecidability is always settled at the next stage up
[Koellner]
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23446
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You can't prove consistency using a weaker theory, but you can use a consistent theory
[Linnebo]
|