Is it possible (even if there is no reason to even want to do this) to expand the hyperreal number line at each infinitesimal to insert a "second layer of infinitesimals"?

Let $\epsilon$ be an infinitesimal hyperreal in the halo of $0$. Can we invent another level of infinitesimals that indicate a second-order-infinitesimal distance from infinitesimals so that relative to these second level infinitesimals, the hyperreal infinitesimals seem like real numbers?

E.g. like let $\delta=(\epsilon+1,\epsilon+\frac12,\epsilon+\frac13,\ldots)$ represent an infinitesimal in the "2nd-level-halo" of $\epsilon$.

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    $\begingroup$ Don't these already exist in the hyperreals? For example, $\epsilon^2$ is ``one layer down'' from $\epsilon$, so you can think of $\epsilon + \epsilon^2 = \epsilon(1 + \epsilon)$ to be in the infinitesimal halo of $\epsilon$. $\endgroup$ – user263190 Jun 11 '18 at 21:36
  • $\begingroup$ Ah, OK, so $\epsilon^2+k\epsilon+k$ is in the halo of $\epsilon$ then? For $k$ a real. $\endgroup$ – jdods Jun 11 '18 at 22:35
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    $\begingroup$ No, once you add $k$ to it, it's no longer near $\epsilon$ (which is close to $0$), because $k + k\epsilon + \epsilon^2$ is close to $k$. $\endgroup$ – user263190 Jun 12 '18 at 4:35
  • $\begingroup$ OK, I think I am kind of understanding. So there is no way to create more infinitesimals that aren't already included in the hyperreals it seems. $\endgroup$ – jdods Jun 12 '18 at 4:41

Indeed it is possible to do so. Terry Tao argued for the advantages of a number system containing multiple levels of infinitesimals. Here the idea is that given a hyperreal system $S={}^\ast\mathbb R$ and a nonzero infinitesimal $\epsilon\in{}^\ast\mathbb R$, we wish to construct a further extension $S\hookrightarrow T$ where $T$ would behave with respect to $S$ as $S$ behaves with respect to $\mathbb R$. Thus, there will be a nonzero infinitesimal $\delta \in T$ that's smaller than "anything that can be expressed in terms of $\epsilon$", where the last phrase requires clarification of course. For Tao's take on this see for instance this post where he talks about levels $\eta_0,\eta_1,\eta_2$.

  • $\begingroup$ Thanks for finding this! So this confirms my intuition. In a sense, we can "increase the density" of the number line by countably adding new levels of infinitesimals. $\endgroup$ – jdods Jun 12 '18 at 11:12
  • $\begingroup$ I am not sure what you mean by "countably". Do you mean to say that you want to have countably many such levels? While in my answer I only mentioned finitely many levels following Tao, in fact one can indeed have countably many levels following Hrbacek; see for example this 2017 publication in Real Analysis Exchange. $\endgroup$ – Mikhail Katz Jun 12 '18 at 11:16
  • $\begingroup$ I assumed the result for having any finite number of levels could imply that we could create a countably infinite number of levels of infinitesimals, but of course that is a leap which probably requires careful reasoning. Thanks for the link. This is probably way over my head, technically. I'm mostly just interested in the intuitive picture for now. My intuition is that at each point of a number line, we can create an "embedded number line of infinitesimals". $\endgroup$ – jdods Jun 12 '18 at 14:53

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