The question is: what is this model? What are its properties? Is there only one such model? Nonetheless, the theorem proves that there is a model (a number system) in which this will be true. This is not true in the real numbers ( R) given by ZFC. In the second expression, the statement says that there is an ‘x’’ (at least one), chosen first, which is between 0 and 1/n for any n. Here, one chooses n first, then one finds the corresponding x. The first statement is true in the real numbers as given in ZFC set theory: for any positive integer n it is possible to find a real number between 1/n and zero, only this real number will depend on n. The possibility to switch “for any' and 'there exists' is crucial. A consequence of this theorem is that if there is a number system in which it is true that for any positive integer n there is a positive number x such that 0< x < 1/n, then there exists an extension of that number system in which it is true that there exists a positive number x such that for any positive integer n we have 0 < x < 1/ n. This theorem is fundamental for the existence of infinitesimals as it proves that it is possible to formalize them. In 1936 Maltsev proved the compactness theorem. We consider here systems where infinitesimals can be shown to exist. Proving or disproving the existence of infinitesimals of the kind used in nonstandard analysis depends on the model and which collection of axioms are used. This is the crucial point: infinitesimal must necessarily mean infinitesimal with respect to some other type of numbers. If epsilon is infinitesimal with respect to a class of numbers, it means that epsilon cannot belong to that class. Infinitesimal is necessarily a relative concept. Neither formulation is wrong, and both give the same results if used correctly.

In the twentieth century, it was found that infinitesimals could, after all, be treated rigorously. It was not until the second half of the nineteenth century that the calculus was given a formal mathematical foundation by Karl Weierstrass and others using the notion of a limit. That step should yield the 'largest' number, but clearly there is no 'last' biggest number. If h is such a number, then what is h/2? Or, if h is indivisible, is it still a number? Also, intuitively, one would require the reciprocal of an infinitesimal to be infinitely large (in modulus) or unlimited. Considering positive numbers, the only way for a number to be less than all numbers would be to be the least positive number. When we consider numbers, the naive definition is clearly flawed: an infinitesimal is a number whose modulus is less than any non-zero positive number. The fundamental problem is that d x is first treated as non-zero (because we divide by it), but later discarded as if it were zero. The use of infinitesimals was attacked as incorrect by Bishop Berkeley in his work The Analyst. This argument, while intuitively appealing, and producing the correct result, is not mathematically rigorous. To find the derivative f′( x) of the function f( x) = x 2, let d x be an infinitesimal. When Newton and Leibniz developed calculus, they made use of infinitesimals.

In India, from the twelfth to the sixteenth century, infinitesimals were discovered for use with differential calculus by Indian mathematician Bhaskara and various Keralese mathematicians. The Archimedean property is the property of an ordered algebraic structure having no nonzero infinitesimals. The first mathematician to make use of infinitesimals was Archimedes (around 250 B.C.E.).

Infinitesimals def