Yes, students have trouble with infinity. And with zero. The great James Propp has written an in-depth essay about conceptual and linguistic issues with infinity and zero.

Concrete examples are always best. Propp offers eight ways in which a student might describe an infinite solution set. Which if any of these do you find acceptable?

1. “There are infinity solutions.”
2. “There are infinite solutions.”
3. “There is/are an infinity of solutions.”
4. “There are infinitely many solutions.”
5. “There is/are an infinitude of solutions.”
6. “There’s an infinite number of solutions.”
7. “There is an infinite amount of solutions.”
8. “The set of solutions is infinite.”

Personally I prefer #4—as does Propp. His comment:

The linguistic divide seems to become entrenched at age twenty or earlier: “infinitely many” sounds correct to people who are math majors but sounds excessively formal to people who are not math majors.

Maybe so. I was never a math major, but as a long-time math teacher I suppose I count as a member of that set.

Then there are surprisingly many (but not infinitely many) related issues such as how to describe the situation in other languages. And of course there’s Cantor’s Hotel and the related thorny issue of there being different sizes of infinity—an issue that I refuse to face in my math teaching until the last week of precalculus.

Anyway, this is a surprisingly rich topic, much too complex for this short blog post, so go read Propp’s entire essay. Please.

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