Is $\textrm{d}x \in \mathbb{R}$?

Question

Is an infinitesimal a real number? Can "abuse of Leibniz's notation" be justified by claiming that an infinitesimal is a real number? If not, what is an infinitesimal?

Answer 1

An infinitesimal is an element of a non-standard model of the Reals (where the existence of this model is given by Compactness and Lowenheim-Skolem theorems; model of uncountable cardinality) that does not exist in the standard Reals, because an infinitesimal does not satisfy the Archimedean property/axiom of the standard Reals.

Answer 2

Infinitesimal quantities are, historically, a less consistent way of thinking of limits. The derivatives and the integrals where useful, but the correct idea of limits wasn't developed until more than hundred years after Newton and Leibniz.

In calculus the idea of limits has replaced the idea of infinitesimals. But of course, $dx$ could be thought of as $\displaystyle \lim_{\Delta x\to 0}\Delta x$.

Only when identifying $dx=0$ it is correct that $dx\in\Bbb R$. But one could study the ring of formal polynomials $\Bbb R[dx]$ (or the field of rational functions $\Bbb R(dx)$) in which the constant terms correspond to the real numbers and here $dx\in \Bbb R[dx]$.