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Single Idea 10118
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
]
Full Idea
First Incompleteness Theorem: If S is a sufficiently powerful formal system, then if S is consistent then S is syntactically incomplete.
Gist of Idea
First Incompleteness: a decent consistent system is syntactically incomplete
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.161
A Reaction
Gödel found a single sentence, effectively saying 'I am unprovable in S', which is neither provable nor refutable in S.
The
40 ideas
from Kurt Gödel
9942
|
Gödel proved the classical relative consistency of the axiom V = L
[Gödel, by Putnam]
|
10868
|
The Continuum Hypothesis is not inconsistent with the axioms of set theory
[Gödel, by Clegg]
|
13517
|
If set theory is consistent, we cannot refute or prove the Continuum Hypothesis
[Gödel, by Hart,WD]
|
18062
|
Set-theory paradoxes are no worse than sense deception in physics
[Gödel]
|
8679
|
We perceive the objects of set theory, just as we perceive with our senses
[Gödel]
|
10271
|
Basic mathematics is related to abstract elements of our empirical ideas
[Gödel]
|
17751
|
Gödel proved the completeness of first order predicate logic in 1930
[Gödel, by Walicki]
|
17835
|
Gödel show that the incompleteness of set theory was a necessity
[Gödel, by Hallett,M]
|
21752
|
Prior to Gödel we thought truth in mathematics consisted in provability
[Gödel, by Quine]
|
17886
|
The limitations of axiomatisation were revealed by the incompleteness theorems
[Gödel, by Koellner]
|
10071
|
Second Incompleteness: nice theories can't prove their own consistency
[Gödel, by Smith,P]
|
19123
|
If soundness can't be proved internally, 'reflection principles' can be added to assert soundness
[Gödel, by Halbach/Leigh]
|
17888
|
The undecidable sentence can be decided at a 'higher' level in the system
[Gödel]
|
10621
|
Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme
[Smith,P on Gödel]
|
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]
|
8747
|
Realists are happy with impredicative definitions, which describe entities in terms of other existing entities
[Gödel, by Shapiro]
|
3192
|
Basic logic can be done by syntax, with no semantics
[Gödel, by Rey]
|
10132
|
There can be no single consistent theory from which all mathematical truths can be derived
[Gödel, by George/Velleman]
|
10041
|
Impredicative Definitions refer to the totality to which the object itself belongs
[Gödel]
|
21716
|
In simple type theory the axiom of Separation is better than Reducibility
[Gödel, by Linsky,B]
|
10035
|
Mathematical Logic is a non-numerical branch of mathematics, and the supreme science
[Gödel]
|
10038
|
A logical system needs a syntactical survey of all possible expressions
[Gödel]
|
10039
|
Some arithmetical problems require assumptions which transcend arithmetic
[Gödel]
|
10042
|
Reference to a totality need not refer to a conjunction of all its elements
[Gödel]
|
10043
|
Mathematical objects are as essential as physical objects are for perception
[Gödel]
|
10045
|
Impredicative definitions are admitted into ordinary mathematics
[Gödel]
|
10046
|
The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers
[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]
|
10620
|
Originally truth was viewed with total suspicion, and only demonstrability was accepted
[Gödel]
|
17883
|
Gödel's Theorems did not refute the claim that all good mathematical questions have answers
[Gödel, by Koellner]
|
9188
|
Gödel proved that first-order logic is complete, and second-order logic incomplete
[Gödel, by Dummett]
|
17892
|
For clear questions posed by reason, reason can also find clear answers
[Gödel]
|