# Definition:Infinitesimal

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## Definition

An **infinitesimal** is a mathematical object $\delta$ resembling a (real) number such that:

- $(1): \quad \delta > 0$
- $(2): \quad \forall x \in \R_{>0}: \delta < x$

That is, an **infinitesimal** is a (strictly) positive real number which is smaller than every other (strictly) positive real number.

## Historical Note

The concept of an **infinitesimal** was initially proposed by Gottfried Wilhelm von Leibniz when inventing calculus.

The logical inconsistencies of this approach, famously criticised by Bishop Berkeley, caused it to be abandoned.

However, the concept has been revived in recent years with the invention of non-standard analysis.

## Sources

- 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $8$: The System of the World: Leibniz - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**infinitesimal**