more from this thinker | more from this text
Full Idea
Second-order arithmetic is categorical - indeed, there is a single formula of second-order logic whose only model is the standard model ω, consisting of just the natural numbers, with all of arithmetic following. It is nevertheless incomplete.
Gist of Idea
Although second-order arithmetic is incomplete, it can fully model normal arithmetic
Source
Stephen Read (Thinking About Logic [1995], Ch.2)
Book Ref
Read,Stephen: 'Thinking About Logic' [OUP 1995], p.47
A Reaction
This is the main reason why second-order logic has a big fan club, despite the logic being incomplete (as well as the arithmetic).
| 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] |