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For $r$ is a real number, I can write $r \in \mathbb{R}$.

For $\varepsilon$ is an infinitesimal, I'd like to write something like $\varepsilon \in something$ Is there a symbol for "the set of infinitesimals"? Or alternatively, a commonly used abbreviation for "infinitesimal"?

For $H$ is an infinite (hyperreal) number, I'd like to write something like $H \in \infty$ Is there a symbol for "the set of infinite hyperreals", or a common abbreviation?

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    $\begingroup$ Assuming there is no standard notation (which is my guess), you could always just make up your own notation. In fact I have a suggestion: because $\omega$ is often used for an infinite ordinal and $\delta$ for an infinitesimal, I'd recommend something like $H_\omega$ and $H_\delta$ (though to be honest, I'm not thrilled with the label $H$ for the hyperreals as that's usually used for the quaternions. I've seen $^*\mathbf R$ used for the hyperreals before). $\endgroup$
    – user137731
    Nov 18, 2016 at 16:23
  • $\begingroup$ @YvesDaoust This is in the context of non-standard analysis. $\endgroup$
    – mhwombat
    Nov 23, 2016 at 14:07
  • $\begingroup$ @mhwombat: I suggest to add that as a keyword. If I am right, the hyperreals also include the infinitesimals,and the set of hyperreals is denoted $^*\mathbb R$, as you say. $\endgroup$
    – user65203
    Nov 23, 2016 at 14:08
  • $\begingroup$ @YvesDaoust Done $\endgroup$
    – mhwombat
    Nov 23, 2016 at 14:08
  • $\begingroup$ For hyperreals, you can use the concept monad or halo. $x$ infinitesimally small can be expressed as $x \in \mathrm{monad}(0)$ or $x \in \mathrm{hal}(0)$. $\endgroup$
    – achille hui
    Nov 23, 2016 at 18:15

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In non-standard analysis, a monad (also called halo) is the set of points infinitesimally close to a given point.

On model for extending real numbers is the hyperreal numbers. The set of hyperreals is usually denoted as ${}^*\mathbb{R}$. Given $x \in {}^*\mathbb{R}$, the monad of $x$ is the set

$$\mathrm{monad}(x) = \{\; y \in {}^*\mathbb{R} : x - y \text{ is infintesimal }\;\}$$

For those $x$ where $|x| < n$ for $n \in \mathbb{N}$, we call $x$ finite (or limited). For such a $x$, there is a unique real number belongs to the monad of $x$. It will be called the standard part of $x$ (also known as shadow of $x$).

To specify a number $x$ is infinitesimally small, one can use the notation $x \in \mathrm{monad}(0)$ or $x \in \mathrm{hal}(0)$.

If one want to go beyond this single use of notations for infinitesimals, I'll suggest one pick a textbook on this topic and stick to it. For example, I use following book as reference

Lectures on the Hyperreals (an introduction to Nonstandard Analysis) by Robert Goldblatt

It uses following notations

  • Hyperreal $b$ is infinitely close to hyperreal $c$, denoted by $b \simeq c$ if $b - c$ is infinitesimal. This define an equivalent relations on ${}^*\mathbb{R}$. The halo of a point $b$ is the $\simeq$-equivalence class $$\mathrm{hal}(b) = \{ \; c \in {}^*\mathbb{R} : b \simeq c \; \}$$

  • Hyperreals $b$ and $c$ are of limited distance apart, denoted by $b \sim c$, if $b - c$ is limited (i.e $|b-c| < n$ for some $n \in \mathbb{N}$). The galaxy of $b$ is the $\sim$-equivalent class $$\mathrm{gal}(b) = \{ \; c \in {}^*\mathbb{R} : b \sim c \; \}$$

  • The standard part of $x$ is denoted by $\mathrm{sh}(x)$.

Your mileage may vary.

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  • $\begingroup$ Errata: I'm pretty sure that definition of "monad" is only correct for limited $x$ (although that incorrect definition does seem to get some publicity) $\endgroup$
    – user14972
    Nov 30, 2016 at 15:47
  • $\begingroup$ @Hurkyl, Up to my knowledge, the term monad is introduced by Abraham Robinsion. I just checked my copy of Robinsons' "Non-standard analysis". In $\S$ 3.2, it says: For any finite real number $a$ in $*R$, we call the uniquely determined standard real number which is infinitely close to $a$ ($r$ in above proof) the standard part of $a$, in symbols $r = st(a)$ or more briefly $r = {}^0 a$. Moreover, given any real number $a$ in $*R$ we call the set of real numbers which are infinitely close to $a$, the monad of $a$, to be denoted by $\mu(a)$... his definition is not limited to limited $a$ $\endgroup$
    – achille hui
    Nov 30, 2016 at 16:00
  • $\begingroup$ Strange, because that seems to conflict with the definition of a monad of a standard point in $p$ a topological space as "the intersection of all standard neighborhoods of $p$", where we would have that each positive unlimited is (after embedding in the extended reals) contained in the monad of $+\infty$. $\endgroup$
    – user14972
    Nov 30, 2016 at 21:07
  • $\begingroup$ @Hurkyl it is possible when people attempt to generalize Robinson's notion of monad to topological space, there are problem if one allow "monad" beyond limited $x$. I've no idea about that. I just quote definition from a known source. $\endgroup$
    – achille hui
    Nov 30, 2016 at 22:25
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For infinite numbers there is a fairly common notation in the context of integers $\mathbb N$ and hyperintegers ${}^\ast\mathbb N$. Namely, a hyperinteger is infinite if it belongs to the set complement $${}^\ast\mathbb N\setminus\mathbb N.$$ This is not particularly elegant but introducing special notation for this set may cause even greater confusion.

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After more research, I have concluded that, as Bye_world suggests, there is no standard notation for the set of infinitesimals. Here are some of the notations I have seen used:

$\mathcal{I}$, $N$, $\mathbb{N}$, $\Delta$.

Also, for "$x$ is an infinitesimal", I have seen the notation $x \approx 0$.

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