Talk:Induction, bounding and least number principles

It would be nice to have a diagram of the relations between the principles in the "Relations" section. --Jordan Mitchell Barrett (talk) 22:29, 13 April 2021 (UTC)[reply]

Just added one. --Jordan Mitchell Barrett (talk) 04:36, 14 April 2021 (UTC)[reply]

The "Nonstandard Models" section is confused

The section claims that the various induction/LEP/bounding schemas can fail in nonstandard models. But this is simply not the case. Indeed, since these nonstandard models are models of Peano Arithmetic, they have to satisfy at least the induction schema, since that schema is part of PA. Moreover, since the schemas are all equivalent over the base theory PA, any nonstandard model must satisfy them all.

The text claims that we can falsify the induction schema in a nonstandard model as follows: we can let phi be any formula true only on the standard part of the model. Then the first half of the induction axiom for phi is satisfied, but the formula phi doesn't hold for all elements of the model, thereby falsifying the induction axiom. The issue with this argument is that there is no such formula. By the overspill lemma, any formula true of every standard element is true of some nonstandard element as well. The overspill lemma is proved precisely by appeal to the induction axiom.

In light of the above, I recommend deleting the section. CanalConditional (talk) 18:38, 30 April 2023 (UTC)[reply]