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Single Idea 10614

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic ]

Full Idea

The concept of truth of sentences in a language cannot be defined in the language. This is the true reason for the existence of undecidable propositions in the formal systems containing arithmetic.

Gist of Idea

The real reason for Incompleteness in arithmetic is inability to define truth in a language

Source

Kurt Gödel (works [1930]), quoted by Peter Smith - Intro to Gödel's Theorems 21.6

Book Ref

Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.182

A Reaction

[from a letter by Gödel] So they key to Incompleteness is Tarski's observations about truth. Highly significant, as I take it.

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]
21752 Prior to Gödel we thought truth in mathematics consisted in provability [Gödel, by Quine]
17835 Gödel show that the incompleteness of set theory was a necessity [Gödel, by Hallett,M]
10621 Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P on Gödel]
17888 The undecidable sentence can be decided at a 'higher' level in the system [Gödel]
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]
10132 There can be no single consistent theory from which all mathematical truths can be derived [Gödel, by George/Velleman]
10072 First Incompleteness: arithmetic must always be incomplete [Gödel, by Smith,P]
9590 Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Gödel, by Nagel/Newman]
3198 Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey]
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]
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]
10046 The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers [Gödel]
10045 Impredicative definitions are admitted into ordinary mathematics [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]
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