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Single Idea 10611

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

Full Idea

The original Gödel construction gives us a sentence that a theory shows is true if and only if it satisfies the condition of being unprovable-in-that-theory.

Gist of Idea

There is a sentence which a theory can show is true iff it is unprovable

Source

report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 20.5

Book Ref

Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.173


The 40 ideas from Kurt Gödel

Gödel proved the classical relative consistency of the axiom V = L [Gödel, by Putnam]
The Continuum Hypothesis is not inconsistent with the axioms of set theory [Gödel, by Clegg]
If set theory is consistent, we cannot refute or prove the Continuum Hypothesis [Gödel, by Hart,WD]
Set-theory paradoxes are no worse than sense deception in physics [Gödel]
We perceive the objects of set theory, just as we perceive with our senses [Gödel]
Basic mathematics is related to abstract elements of our empirical ideas [Gödel]
Gödel proved the completeness of first order predicate logic in 1930 [Gödel, by Walicki]
Prior to Gödel we thought truth in mathematics consisted in provability [Gödel, by Quine]
Gödel show that the incompleteness of set theory was a necessity [Gödel, by Hallett,M]
The limitations of axiomatisation were revealed by the incompleteness theorems [Gödel, by Koellner]
Second Incompleteness: nice theories can't prove their own consistency [Gödel, by Smith,P]
If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Gödel, by Halbach/Leigh]
Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P on Gödel]
The undecidable sentence can be decided at a 'higher' level in the system [Gödel]
There can be no single consistent theory from which all mathematical truths can be derived [Gödel, by George/Velleman]
Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey]
First Incompleteness: arithmetic must always be incomplete [Gödel, by Smith,P]
Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Gödel, by Nagel/Newman]
Gödel's Second says that semantic consequence outruns provability [Gödel, by Hanna]
First Incompleteness: a decent consistent system is syntactically incomplete [Gödel, by George/Velleman]
Second Incompleteness: a decent consistent system can't prove its own consistency [Gödel, by George/Velleman]
There is a sentence which a theory can show is true iff it is unprovable [Gödel, by Smith,P]
'This system can't prove this statement' makes it unprovable either way [Gödel, by Clegg]
Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro]
Basic logic can be done by syntax, with no semantics [Gödel, by Rey]
Impredicative Definitions refer to the totality to which the object itself belongs [Gödel]
In simple type theory the axiom of Separation is better than Reducibility [Gödel, by Linsky,B]
Mathematical Logic is a non-numerical branch of mathematics, and the supreme science [Gödel]
A logical system needs a syntactical survey of all possible expressions [Gödel]
Some arithmetical problems require assumptions which transcend arithmetic [Gödel]
Reference to a totality need not refer to a conjunction of all its elements [Gödel]
Mathematical objects are as essential as physical objects are for perception [Gödel]
The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers [Gödel]
Impredicative definitions are admitted into ordinary mathematics [Gödel]
Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable [Gödel, by Koellner]
The real reason for Incompleteness in arithmetic is inability to define truth in a language [Gödel]
Originally truth was viewed with total suspicion, and only demonstrability was accepted [Gödel]
Gödel's Theorems did not refute the claim that all good mathematical questions have answers [Gödel, by Koellner]
Gödel proved that first-order logic is complete, and second-order logic incomplete [Gödel, by Dummett]
For clear questions posed by reason, reason can also find clear answers [Gödel]
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