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> To rule out such models - and therefore prove that every natural number is finite - you need the second-order induction axiom. Sorry, I don't see why second-
by lisper 2y ago
> To rule out such models - and therefore prove that every natural number is finite - you need the second-order induction axiom.
Sorry, I don't see why second-order induction is required. AFAICT all you need is a first-order induction axiom instantiated on the predicated FINITE.
- Tainnor 2y agoI don't know why I wrote an entire proof outlining why first-order PA cannot rule out infinite numbers, if you just ignore it and insist on your misconception. The set of finite numbers is undefinable in the language of first-order PA, which is why first-order induction isn't enough. Read a textbook, these are well-known facts. Here's a free one that is quite good, chapter 3 is relevant: https://milneopentextbooks.org/a-friendly-introduction-to-mathematical-logic/ https://milneopentextbooks.org/a-friendly-introduction-to-ma...