I'm not a big fan of using nonstandard analysis for this. We're assuming the existence of arbitrary answers that we cannot ever produce.<p>For example, which function is eventually larger than the other?<p><pre><code> (1 + sin(x)) * e^x + x
(1 + cos(x)) * e^x + x
</code></pre>
In the ultrafilter, one almost certainly will be larger. In fact the ratio of the two will, asymptotically, approach a specific limit. Which one is larger? What is the ratio? That entirely depends on the ultrafilter.<p>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.