Comment by fn-mote

>> If for all n we can prove P(n), then `∀n P(n)` should be an acceptable proposition

> But how can you prove that P(n) for all n without induction?

Just to be clear to all readers, the axiom of COUNTABLE choice is uncontroversial. Nobody is disturbed by induction.

The issue it that when you allow UNCOUNTABLE choice - choices being made for all real numbers (in a non-algorithmic way, I believe - so not a simple formula) - there are some unpleasant consequences.