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6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / e. Peano arithmetic 2nd-order

[Dedekind-Peano axioms which also refer to properties]

9 ideas
Many concepts can only be expressed by second-order logic [Boolos]
Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro]
Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price]
A single second-order sentence validates all of arithmetic - but this can't be proved axiomatically [Sider]
Although second-order arithmetic is incomplete, it can fully model normal arithmetic [Read]
Second-order arithmetic covers all properties, ensuring categoricity [Read]
Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman]
A plural language gives a single comprehensive induction axiom for arithmetic [Hossack]
Categoricity implies that Dedekind has characterised the numbers, because it has one domain [Rumfitt]