Infinitesimalis

Infinitesimalis is a term used in some mathematical discussions to denote a class of infinitesimal elements that inhabit an extended number system built to model quantities smaller than any real number. It is not a standard term in mainstream mathematics, but appears in a range of speculative or expository contexts to emphasize certain structural aspects of infinitesimals.

Etymology derives the name from infinitesimal, with the suffix -is to indicate a collection or class; it echoes the Latin plural form often used in mathematical terminology.

Mathematical background: In nonstandard analysis, infinitesimals are positive numbers smaller than any real number; infinitesimalis may refer to a distinguished subset with particular algebraic or order properties. Examples within extended systems include dual numbers with nilpotent infinitesimals (such as ε, where ε^2 = 0) and hyperreal fields containing infinitesimals of varying magnitudes. Infinitesimalis is sometimes used to discuss a uniform class of infinitesimals across different models or constructions.

Variants and interpretations: Some authors treat infinitesimalis as nilpotent, others as non-nilpotent; some frameworks require a standard part function that maps infinitesimalis to real numbers. In smooth infinitesimal analysis, nilpotent infinitesimals are used to define differentiation axiomatically, illustrating alternative foundations for calculus.

Applications: The concept appears primarily in expository contexts to illustrate calculus foundations, differential geometry, and infinitesimal modeling. It is not widely adopted as a formal theory, but serves as a useful pedagogical device and a way to discuss the intuition behind infinitesimals in various mathematical frameworks.