For example, which function is eventually larger than the other?
(1 + sin(x)) * e^x + x
(1 + cos(x)) * e^x + x
Which means that we can accept the illusionary simplicity of his axiom about every predicate P(N), and it will remain simple right until we try to get a concrete and useful answer out of it.
That still doesn't resolve which one is larger though.
Look for the comment in the article, after passing to a subsequence if necessary. The ultrafilter produces the necessary subsequence for any question that you ask, and will do so in such a way as to produce logically consistent answers for any combination of questions that you choose.
That is why the ultrafilter axiom is a weak version of choice. Take the set of possible yes/no questions that we can ask as predicates, such that each answer shows up infinitely often. The ultrafilter results in an arbitrary yet consistent set of choices of yes/no for each predicate.
O is a total order, but functions aren't in any way a total order, so what's the point?
For instance, your question happens to be equivalent to asking whether sin(omega) > cos(omega), and thus if tan(omega) > 1. This is true iff the fractional part the hyperreal number omega/(2*pi) is between 1/8 and 5/8. Thus we have reduced the asymptotic statement to a question about an arithmetical property of one particular hyperreal number.
Choosing an ultrafilter basically involves simultaneously determining all properties of omega. There are different ultrafilters, each providing a different coherent "universe" which decides all possible predicates in a coherent way. That this is possible (with the axiom of choice) is highly interesting. However, it doesn't seem necessary for asymptotic analysis.
Of course, if there is some "canonical" or "most natural" ultrafilter to choose from, with some magical property universally deemed important, then it would settle your question and all such questions in a natural way.
People who look at asymptotic growth are interested in what happens for all, or occasionally almost all, large n. The possibility of this kind of total order is irrelevant, and therefore uninteresting to people who are interested in that. What Tao is doing is mathematics, but not mathematics of a kind that I, personally, like.
The total order on functions is not an end-goal in itself, but a step in a proof which provides useful results.
That's what mathematics is about. You work with something, then you work with something else. When you count kittens, you can't pet the numbers anymore, but the corresponding integers are still useful.
In elementary Calculus, it really does help people's intuition. In fact it allowed us to formalize a lot of the intuitive arguments through which Calculus was originally built. In analysis, it has helped at least some people's intuition. See, for example, Robinson and Bernstein's proof of the invariant problem. However most people in analysis have found that it isn't that hard to translate the NSA version of such proofs into more familiar terminology, and they don't find the NSA version to help their intuition.
When we go as far afield as the asymptotic growth of functions, I don't see our intuition being helped much by NSA. I could be wrong - I would have been on the wrong side of the importance of oracles in cryptography on somewhat similar intuitions - but it remains my impression.
singularity2001•9mo ago
Here is the axiomatic approach in Julia and Lean https://github.com/pannous/hyper-lean