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Single Idea 19123

[filed under theme 5. Theory of Logic / K. Features of Logics / 3. Soundness ]

Full Idea

Gödel showed PA cannot be proved consistent from with PA. But 'reflection principles' can be added, which are axioms partially expressing the soundness of PA, by asserting what is provable. A Global Reflection Principle asserts full soundness.

Gist of Idea

If soundness can't be proved internally, 'reflection principles' can be added to assert soundness

Source

report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Halbach,V/Leigh,G.E. - Axiomatic Theories of Truth (2013 ver) 1.2

Book Ref

'Stanford Online Encyclopaedia of Philosophy', ed/tr. Stanford University [plato.stanford.edu], p.3

A Reaction

The authors point out that this needs a truth predicate within the language, so disquotational truth won't do, and there is a motivation for an axiomatic theory of truth.

Related Idea

Idea 19124 A natural theory of truth plays the role of reflection principles, establishing arithmetic's soundness [Halbach/Leigh]

The 18 ideas from 'On Formally Undecidable Propositions'

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]
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]
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]
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]