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Single Idea 15653

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic ]

Full Idea

Peano Arithmetic cannot derive its own consistency from within itself. But it can be strengthened by adding this consistency statement or by stronger axioms (particularly ones partially expressing soundness). These are known as Reflexion Principles.

Gist of Idea

We can add Reflexion Principles to Peano Arithmetic, which assert its consistency or soundness

Source

comment on Giuseppe Peano (Principles of Arithmetic, by a new method [1889], 1.2) by Volker Halbach - Axiomatic Theories of Truth (2005 ver) 1.2

Book Ref

'Stanford Online Encyclopaedia of Philosophy', ed/tr. Stanford University [plato.stanford.edu], p.3

The 5 ideas from 'Principles of Arithmetic, by a new method'

15653 We can add Reflexion Principles to Peano Arithmetic, which assert its consistency or soundness [Halbach on Peano]
13949 All models of Peano axioms are isomorphic, so the models all seem equally good for natural numbers [Cartwright,R on Peano]
18113 PA concerns any entities which satisfy the axioms [Peano, by Bostock]
17634 Peano axioms not only support arithmetic, but are also fairly obvious [Peano, by Russell]
17635 Arithmetic can have even simpler logical premises than the Peano Axioms [Russell on Peano]