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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]
| 19123 | If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Gödel, by Halbach/Leigh] |
| 9719 | A proof theory is 'sound' if its valid inferences entail semantic validity [Enderton] |
| 10765 | Soundness would seem to be an essential requirement of a proof procedure [Tharp] |
| 10070 | If everything that a theory proves is true, then it is 'sound' [Smith,P] |
| 10086 | Soundness is true axioms and a truth-preserving proof system [Smith,P] |
| 10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P] |
| 13635 | 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence [Shapiro] |
| 10120 | Soundness is a semantic property, unlike the purely syntactic property of consistency [George/Velleman] |
| 18757 | Soundness theorems are uninformative, because they rely on soundness in their proofs [McGee] |
| 16342 | You cannot just say all of Peano arithmetic is true, as 'true' isn't part of the system [Halbach] |
| 16341 | Normally we only endorse a theory if we believe it to be sound [Halbach] |
| 16344 | Soundness must involve truth; the soundness of PA certainly needs it [Halbach] |