Can such definition of infinitesimals hold?

$$\mathrm{d} x :=a:(a>0 \;\And\; \forall b \in \mathbb{R}^+\backslash \{ a \}\;(a<b))$$

And, if the above definiton works, then obviously

$$\mathrm{d} f(x) := f(x+{\mathrm{d} x})-f(x)$$

This definiton is basically a try not to delve into the realms of non-standard analysis and other sophisticated branches of mathematics, which seems redundant for me for defining such an intuitive concept.

  • $\begingroup$ You don't specify where $a$ comes from: is $a\in \mathbb{R}$? In that case you can easily see that the only number satisfying that property would be zero, which you have excluded with the simbol $>$. $\endgroup$ – pppqqq Jun 17 '13 at 9:49
  • $\begingroup$ I'm no mathematician but if I read this correctly, you're trying to assign $a$ a value? there's no such $a$ because if $a\in R$ (How is the $\lt$ defined?) then $b=a/2$ would be a problem. $\endgroup$ – Guest 86 Jun 17 '13 at 9:50
  • $\begingroup$ Look up the traditional notation for universal quantifiers. $\endgroup$ – Constantine Jun 17 '13 at 9:54
  • $\begingroup$ The sophistication of non-standard analysis is mainly in the foundations -- e.g. its analog of "peano axioms -> construct integers -> construct rationals -> construct reals", although the simplicity of "hyperrationals -> reals" is a minor mitigating factor. Actually using non-standard analysis to do calculus is much less sophisticated, and mainly involves recognizing when a statement, object, or idea is internal or not... which isn't much more than noticing when something involves comparing the standard and the non-standard models. $\endgroup$ – Hurkyl Apr 10 '14 at 17:13

The original problem with the infinitesimal approach to analysis/calculus was the existence of these objects. If $a = \mathrm{d}x$ were a real real number, then so would $\frac{a}{2}$. As $a > 0$, basic arithmetic says that $0 < \frac{a}{2} < a$, which means that $a$ cannot have the defining property of being an infinitesimal!

What the sophisticated machinery of Abraham Robinson did was establish a logically consistent and rigourous footing for the intuitive notion of an infinitesimal. After this was done, one can do non-standard analysis/calculus without worrying at all about the inner details. (Much like most mathematicians think of the real number $\pi$ as a single discrete object, when it is actually an infinite set of rational numbers! Or maybe it is an equivalence class of Cauchy sequences of rational numbers. I forget.)

  • 3
    $\begingroup$ Indeed, H. Jerome Keisler has a freshman calculus book using infinitesimals which does "non-standard analysis/calculus without worrying about all the inner details." It's very good in my opinion and it's freely available at the link. $\endgroup$ – joeA Jun 17 '13 at 11:09

If $a\in\mathbb{R}$, your definition of infinitesimal is empty, since $\mathbb{R}$ is an (ordered) Archimedean field.


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