![]() ![]() Other mathematical systems exist which include infinitesimals, including nonstandard analysis and the surreal numbers. However, there are also models that include invertible infinitesimals. In typical models of smooth infinitesimal analysis, the infinitesimals are not invertible, and therefore the theory does not contain infinite numbers. However, there are plenty of x, namely the infinitesimals, such that neither x = 0 nor x ≠ 0 holds, so the function is not defined on the real numbers. If the law of the excluded middle held, then this would be a fully defined, discontinuous function. One can see that the law of excluded middle cannot hold from the following basic theorem (again, understood in the context of a theory of smooth infinitesimal analysis):Įvery function whose domain is R, the real numbers, is continuous and infinitely differentiable.ĭespite this fact, one could attempt to define a discontinuous function f( x) by specifying that f( x) = 1 for x = 0, and f( x) = 0 for x ≠ 0. In particular, in a theory of smooth infinitesimal analysis one can prove for all infinitesimals ε, NOT ( ε ≠ 0) yet it is provably false that all infinitesimals are equal to zero. This approach departs from the classical logic used in conventional mathematics by denying the law of the excluded middle, e.g., NOT ( a ≠ b) does not imply a = b. The nilsquare or nilpotent infinitesimals are numbers ε where ε² = 0 is true, but ε = 0 need not be true at the same time. As a theory, it is a subset of synthetic differential geometry. Lawvere and employing the methods of category theory, it views all functions as being continuous and incapable of being expressed in terms of discrete entities. Smooth infinitesimal analysis is a modern reformulation of the calculus in terms of infinitesimals. Modern reformulation of the calculus in terms of infinitesimals
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