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Single Idea 16342

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

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]

The 12 ideas with the same theme [whether all formal deductions always lead to truth]:

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]