Page "Non-standard calculus" Paragraph 10

The hyperreals can be constructed in the framework of Zermelo-Fraenkel set theory, the standard axiomatisation of set theory used elsewhere in mathematics.

To give an intuitive idea for the hyperreal approach, note that, naively speaking, non-standard analysis postulates the existence of positive numbers ε which are infinitely small, meaning that ε is smaller than any standard positive real, yet greater than zero.

Every real number x is surrounded by an infinitesimal " cloud " of hyperreal numbers infinitely close to it.

To define the derivative of f at a standard real number x in this approach, one no longer needs an infinite limiting process as in standard calculus.

Instead, one sets