# Doubly-hyper-reals? Can we include another level of infinitesimals?

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 $\epsilon$.

• 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 $\epsilon$. – user263190 Jun 11 '18 at 21:36
• Ah, OK, so $\epsilon^2+k\epsilon+k$ is in the halo of $\epsilon$ then? For $k$ a real. – jdods Jun 11 '18 at 22:35
• 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 $k$. – user263190 Jun 12 '18 at 4:35
• 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. – 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 $\eta_0,\eta_1,\eta_2$.