There is a set of related topics in a freshman calc course that includes the completeness axiom for the reals, the intermediate value theorem, extreme value theorem, Rolle's theorem, and mean value theorem.
How can one interest a freshman calc student in these topics, when they almost certainly have zero interest in foundational issues?
This material seems to me to have a fairly tenuous connection to the mainstream of a freshman calc course. I'm not an expert in the history of mathematics, but it seems that Dedekind cuts and such were not invented until about 1860. That leaves a period of about 150 years during which the world's greatest mathematicians happily did calculus without feeling a need for these foundational issues to be addressed. They would probably have felt that a result like the extreme value theorem was geometrically obvious. (Some of the theorems were stated earlier, e.g., Rolle's theorem dates to 1691, but I don't see how Rolle could have given a meaningful proof 150 years before the arithmetization of the reals.)
Further adding to the difficulties in selling this to a class of students are the facts that:
These theorems are not particularly useful in solving problems (because in most problems where they were needed, the theorems' claims would have been intuitively obvious anyway).
In a freshman calculus course we don't really attempt to present this material in detail. E.g., textbooks tend to omit most of the proofs.
(Just to put this in perspective, I recently asked a class of students who had all already had a year of calculus to describe in general terms how the derivative was defined -- I wasn't asking them to regurgitate the definition, just to describe its flavor. Despite a lot of poking, prodding, and hinting, not a single student volunteered that it was defined using a limit.)
The best idea I've been able to come up with so far is just to give the class a series of tasks in which they come up with counterexamples to the various theorems when one of the assumptions is omitted, e.g., give an example of a function that is continuous everywhere in its domain but that never attains a maximum value. This might at least help to convince them that the results make sense and have a geometrically visualizable interpretation.