The grader may not know how to grade a proof using nonstandard calculus, as well (which is reasonable, since it invokes a whole host of concepts not used in standard calculus). If the students never take another formal calculus course, this is not a concern, but if they do, all the proof methods they learned in nonstandard calculus will no longer be taught in light of new material. The main issue I see is when they take Calculus in the future.Both standard and nonstandard calculus generalize to other fields, such as topology, so that is not an issue.You still have to do the work, but the work is not separate from the intuition. They are also superior to informal proofs using infinitesimals (like scientists sometimes use) since it never proves an contradictions. Infinity is no longer just an idea, its a number. In nonstandard calculus, most of the intuitions are included literally in the proof. Then you translate those intuitions into a series of epsilon delta definitions, with many inter-dependencies, never including your intuitions in your proof, just their translation. With standard calculus, you usually motivate theorems with ideas like infinity, or being infinitely close, or dividing infinitesimal numbers. Although the proofs are equally as long, they are much more intuitive in nonstandard calculus than in standard calculus.All in all, proofs in both are about equally as long. You can not even build a hyperreal number calculator. This however is balanced by the fact that it is much more difficult to do arithmetic with hyperreal numbers than the real numbers. Limits, continuity, and uniform continuity all have a quantifier complexity of $1$, whereas in standard calculus they have 3 or 4. Definitions in nonstandard calculus involve much fewer quantifiers than those in standard analysis.The main difference is how they got about proving them. In terms of provable statements, the only difference is that nonstandard analysis proves some things about infinite and infinitesimal numbers that standard analysis is not concerned with (since it does not use such number), and even those can be translated into equivalent statements about real numbers in standard analysis. By that I mean that anything you can prove in analysis can be proven in nonstandard analysis, and anything you can disproof is analysis can be proven in standard analysis, so there will never be a contradictory result.
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