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Full Idea
One cannot just accept that all the theorems of Peano arithmetic are true when one accepts Peano arithmetic as the notion of truth is not available in the language of arithmetic.
Gist of Idea
You cannot just say all of Peano arithmetic is true, as 'true' isn't part of the system
Source
Volker Halbach (Axiomatic Theories of Truth [2011], 22.1)
Book Ref
Halbach,Volker: 'Axiomatic Theories of Truth' [CUP 2011], p.322
A Reaction
This is given as the reason why Kreisel and Levy (1968) introduced 'reflection principles', which allow you to assert whatever has been proved (with no mention of truth). (I think. The waters are closing over my head).
Related Idea
Idea 16343 The global reflection principle seems to express the soundness of Peano Arithmetic [Halbach]
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] |