more from Volker Halbach

### Single Idea 16343

#### [catalogued under 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic]

Full Idea

The global reflection principle ∀x(Sent(x) ∧ Bew[PA](x) → Tx) …seems to be the full statement of the soundness claim for Peano arithmetic, as it expresses that all theorems of Peano arithmetic are true.

Clarification

Bew[PA] means provable in Peano arithmetic. T means 'true'

Gist of Idea

The global reflection principle seems to express the soundness of Peano Arithmetic

Source

Volker Halbach (Axiomatic Theories of Truth [2011], 22.1)

Book Reference

Halbach,Volker: 'Axiomatic Theories of Truth' [CUP 2011], p.323

A Reaction

That is, an extra principle must be introduced to express the soundness. PA is, of course, not complete.

Related Idea

Idea 16342
You cannot just say all of Peano arithmetic is true, as 'true' isn't part of the system **[Halbach]**