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Single Idea 17885
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
]
Full Idea
Eventually Gödel ...expressed the hope that there might be a generalised completeness theorem according to which there are no absolutely undecidable sentences.
Gist of Idea
Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable
Source
report of Kurt Gödel (works [1930]) by Peter Koellner - On the Question of Absolute Undecidability Intro
Book Ref
-: 'Philosophia Mathematica' [-], p.2
A Reaction
This comes as a bit of a shock to those who associate him with the inherent undecidability of reality.
Related Idea
Idea 17883
Gödel's Theorems did not refute the claim that all good mathematical questions have answers [Gödel, by Koellner]
The
21 ideas
with the same theme
[discovery that axioms can't prove all truths of arithmetic]:
15653
|
We can add Reflexion Principles to Peano Arithmetic, which assert its consistency or soundness
[Halbach on Peano]
|
10072
|
First Incompleteness: arithmetic must always be incomplete
[Gödel, by Smith,P]
|
3198
|
Gödel showed that arithmetic is either incomplete or inconsistent
[Gödel, by Rey]
|
9590
|
Arithmetical truth cannot be fully and formally derived from axioms and inference rules
[Gödel, by Nagel/Newman]
|
11069
|
Gödel's Second says that semantic consequence outruns provability
[Gödel, by Hanna]
|
10118
|
First Incompleteness: a decent consistent system is syntactically incomplete
[Gödel, by George/Velleman]
|
10122
|
Second Incompleteness: a decent consistent system can't prove its own consistency
[Gödel, by George/Velleman]
|
10611
|
There is a sentence which a theory can show is true iff it is unprovable
[Gödel, by Smith,P]
|
10867
|
'This system can't prove this statement' makes it unprovable either way
[Gödel, by Clegg]
|
10039
|
Some arithmetical problems require assumptions which transcend arithmetic
[Gödel]
|
17885
|
Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable
[Gödel, by Koellner]
|
10614
|
The real reason for Incompleteness in arithmetic is inability to define truth in a language
[Gödel]
|
10067
|
Gentzen proved the consistency of arithmetic from assumptions beyond arithmetic
[Gentzen, by Musgrave]
|
10554
|
Intuitionists find the Incompleteness Theorem unsurprising, since proof is intuitive, not formal
[Dummett]
|
10604
|
Incompleteness results in arithmetic from combining addition and successor with multiplication
[Smith,P]
|
10848
|
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
|
The incompletability of formal arithmetic reveals that logic also cannot be completely characterized
[Hale/Wright]
|
10128
|
The Incompleteness proofs use arithmetic to talk about formal arithmetic
[George/Velleman]
|
17891
|
Arithmetical undecidability is always settled at the next stage up
[Koellner]
|
23446
|
You can't prove consistency using a weaker theory, but you can use a consistent theory
[Linnebo]
|