I think DanielBMarkham is being rather loose with terminology here (or perhaps talking about something other than the Riemann sphere, but it sounds like that's what he's talking about).
The problem isn't that you "can't define integers" when you introduce an infinity. The problem is (as the article I linked to says) the resulting system does not form a field (i.e. the usual arithmetic operations don't behave like you would expect anymore). Follow the link to fields if you want to know more.
Oh, I thought he was saying something like you could have real numbers with division by zero as far as you didn't want to have a special class of numbers, the integer numbers. So I was very curious about that, as I studied fields and all that a long while ago and that sounded like a rare concept.
"Mostly" true, but if you accept this statement as-is, it's a little misleading.
The integers (with standard addition/multiplication operators) are indeed a ring, but a ring is a very loose definition that does not reveal say too much about the integers themselves. There are several stronger algebraic structures than integers that don't have division.
The integers themselves are an example of a general algebraic structure called a "domain," (or, more commonly, "integral domain") which is a commutative ring (that is, commutative under multiplication as well), has distinct additive and multiplicative identities, and has the property that if a and b are integers and a*b = 0, then either a or b must be zero.
When mathematicians in the past were studying divisibility in the integers, they have generally studied domains, because they essentially isolate the division property among different operations. Even an integral domain is not the strongest structure you can place on integers. There are even more specific subclasses of integral domains that the integers fall under, but that's best left for a class on algebraic structures and not a comment on HN.
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