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Full Idea
Soundness is normally a matter of having true axioms and a truth-preserving proof system.
Gist of Idea
Soundness is true axioms and a truth-preserving proof system
Source
Peter Smith (Intro to Gödel's Theorems [2007], 03.4)
Book Ref
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.24
A Reaction
The only exception I can think of is if a theory consisted of nothing but the axioms.
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] |
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] |
16342 | You cannot just say all of Peano arithmetic is true, as 'true' isn't part of the system [Halbach] |