This looks like the most in-depth resource on the topic that I've seen so far; thanks for adding it! One of the reasons that I'm partial to the hyperreals is because it's such a natural thing, in the context of mathematical history, to extend the number system when that system isn't expressive enough to solve the problems we want to solve. The limit-based approach seems clumsy in comparison.
That same idea applies very nicely to other ideas of math too. Take Konig's Lemma (every infinite, locally finite, connected graph has an infinite path) as an example. The nonstandard proof goes something like:
1. For every natural, you can find a path of that length.
2. Therefore (nonstandard chicanery), for every hypernatural you can find a hyperpath of that hyperlength. Pick one for some infinite hypernatural.
3. Restricting that hyperpath to the original graph yields the infinite normal path you were looking for.
Whole problems melt away entirely as soon as you don't have to worry about clumsy "limit-based" approaches.
Comments
This looks like the most in-depth resource on the topic that I've seen so far; thanks for adding it! One of the reasons that I'm partial to the hyperreals is because it's such a natural thing, in the context of mathematical history, to extend the number system when that system isn't expressive enough to solve the problems we want to solve. The limit-based approach seems clumsy in comparison.
That same idea applies very nicely to other ideas of math too. Take Konig's Lemma (every infinite, locally finite, connected graph has an infinite path) as an example. The nonstandard proof goes something like:
1. For every natural, you can find a path of that length.
2. Therefore (nonstandard chicanery), for every hypernatural you can find a hyperpath of that hyperlength. Pick one for some infinite hypernatural.
3. Restricting that hyperpath to the original graph yields the infinite normal path you were looking for.
Whole problems melt away entirely as soon as you don't have to worry about clumsy "limit-based" approaches.