He does not prove that this is a useful representation, only that given his own axioms, this can be considered mathematically correct.
Very practical issues with 1/0 == 0:
- This result is counter-intuitive, took building a custom fields and responding to the incredulity of all. - The main reason this is counter-intuitive is not mentionned in the post: division is no longer monotonous.
1/0.5 = 2 1/0.25 = 4 1/0.0001 = 10000 1/0 = 0
This is calling for an Ariane 5-type crash because an underflow error caused a value to suddenly fall from 10e6 to 0.