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Comment by grannyg00se | Hacker News Reader

"The further I got through grad school, the more I said "unbounded" rather than "infinite"."

I think there's a huge difference between simply saying "unbounded" compared to going through that detailed explanation you just gave. In the latter case, it's very clear what the difference is. But without the explanation I'd guess that people will think something like this:

    unbounded -> without bound -> infinite

In other words, without the explanation "unbounded" becomes a synonym for "infinite".

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"Unbounded" means "any finite number, no matter how large." For many people, this might be what they think of when they hear "infinite," but in mathematics, "infinity" is an entirely different concept.

Even in math "infinity" often means "unbounded". For example, in real analysis if we have a sequence x_n, we write lim x_n = ∞ to mean "given any real number alpha, we can find a natural number K, such that x_n >= alpha for all n > K".

As I mention in another comment, I am specifically referring to the fact that when we computer scientists or discrete mathematicians use the term in proofs, we were more often interested in the "unbounded"ness than something with "actual" infinite cardinality... the fact that I understand the difference is precisely why I tended to start saying unbounded more often. Using infinity to mean unbounded is really too powerful, and a proof should use the least power necessary. (It was only an intuitive feeling at the time; I have a much better understanding of that fact now.)

When infinity is the relevant concept, use it.

This reminds me of the distinction between random and arbitrary. Seems to trip people up sometimes.

The relevant distinction isn't between "bounded" and "finite", but between the size of a container and the size of its elements.

The difference between "bounded" and "infinite" is the difference between "measure" and "cardinality". That is , "bounded" refers to a notion of distance, while "finite" refers to a measure of count. There are bounded infinities (such as the set of all rational or numbers between 0 and 1), but there are no unbounded finities.

A container that contains finite objects of all sizes must be an infinite/unbounded container.

Depending on the context unbounded and infinite may mean very different things, for example the surface of a sphere being unbounded but finite.

A sphere is bounded but infinite*

http://en.wikipedia.org/wiki/Compact_space

*Infnite cardinality of points, not infinite in measure.

That's why I didn't simply drop the word infinite. There are cases where it is appropriate. What is the cardinality of the natural numbers? Infinite... "unbounded" is definitely not right there. But when we use infinity in proofs, we are more often using it to mean "unbounded" rather than truly a "manifested infinity", to use a sloppy phrase; we're using it to get rid of the nasty edge case, not because we expect a real implementation of whatever the proof is proving to actually require infinite RAM.

Also, by "we", I should clarify I mean discrete mathematicians and computer scientists, who are usually dealing in the discrete case. In the continuous case, infinities come up "for real" more often.

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