Look you're coming at this from a mathematics perspective and worrying about every single detail. There's value in this, obviously, but it's unnecessary in practical use, in the same way that I don't have to explain Dedekind cuts to kids before they start to work with real numbers. Nor do I explain measure theory to beginners before they start integrating stuff.

> What if you picked N^2? or 2^N? or some upper limit set whose existence is stronger than choice?

I'm not sure what you mean about picking N to be an upper limit set. N is a hyperreal here, not an ordinal. There aren't really set theoretic difficulties, you can easily construct a model of the hyperreals in ZFC.

It doesn't matter what representative you pick for your Heaviside function - so long as it differs pointwise from the standard Heaviside function by infinitesmals you will get a delta function by differentiating it in NSA. And continuing to differentiate it will give you the higher multiple moments. This is what I meant in my previous response.

It's useful to have the choice because depending on what you want to model you can have non-standard functions that "go to infinity" twice as fast as other functions, for instance. Taking the equivalence class destroys that information, which is sometimes useful, and sometimes not. If all you care about is a small computation you can just pick a representative and move on, I don't think it's a big deal.

Anyway, I'm going to leave this discussion now - we're not really bickering over anything important to my mind. Use the tool you like! Personally I'm having fun playing with NSA right now. Thanks for your time.