On 15 Jan., 21:06, Virgil <vir...@ligriv.com> wrote:
> The induction procedure exhausts both the first natural and also the > successor of every natural, so what naturals does it fail to exhaust?
If the induction procedure was sufficient to establish an actually infinite set, ZF would not need the axiom of infinity.
The induction procedure establishes that there is no actually infinite set by proving that the number of elements up to every natural is finite. As this holds always, the naturals are potentially infinite but we never gather a number of naturals that is larger than any natural. We get it only by reversion of quantification. A simple mistake:
For every n there is m > n has been confused to There is an m larger than every n.