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Full Idea
Second-order arithmetic can rule out the non-standard models (with non-standard numbers). Its induction axiom crucially refers to 'any' property, which gives the needed categoricity for the models.
Clarification
'Categoricity' is when all the models are isomorphic to one another
Gist of Idea
Second-order arithmetic covers all properties, ensuring categoricity
Source
Stephen Read (Thinking About Logic [1995], Ch.2)
Book Ref
Read,Stephen: 'Thinking About Logic' [OUP 1995], p.49
Related Idea
Idea 16321 The compactness theorem can prove nonstandard models of PA [Halbach]
| 18841 | Categoricity implies that Dedekind has characterised the numbers, because it has one domain [Rumfitt on Dedekind] |
| 10833 | Many concepts can only be expressed by second-order logic [Boolos] |
| 10294 | Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro] |
| 10164 | Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price] |
| 13701 | A single second-order sentence validates all of arithmetic - but this can't be proved axiomatically [Sider] |
| 10979 | Although second-order arithmetic is incomplete, it can fully model normal arithmetic [Read] |
| 10980 | Second-order arithmetic covers all properties, ensuring categoricity [Read] |
| 17899 | Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman] |
| 10674 | A plural language gives a single comprehensive induction axiom for arithmetic [Hossack] |