infinitesimals in the real number system

When I raise the same argument about an infinite set, you can’t reply that you can always make the set bigger if I say add an element, you reply that the sets are still the same size (cardinality).

If I deny you can, you reply that since you can reduce it to zero and the function is continuous, you necessarily can reduce any given quantity-precluding actual infinities. The role of zero as a boundary accounts for why it’s obvious you can always reduce a quantity. For however small a quantity you choose, it’s obvious that you can make it yet smaller. But when you multiply an infinitesimal by infinity, you sometimes get a finite quantity: the points on a line are of infinitesimal dimension, in that they occupy no space (as if they were zero duration), yet compose lines finite in extent.įew advocate actual infinitesimals because an actually existing infinitesimal is indistinguishable from zero. When you add an infinitesimal to a real number, it’s like adding zero.

Therefore, actual infinities can’t exist.Īlthough Cantor, who invented the mathematics of transfinite numbers, rejected infinitesimals, mathematicians have continued to develop analyses based on them, as mathematically legitimate as are transfinite numbers, but few philosophers try to justify actual infinitesimals, which have some of the characteristics of zero and some characteristics of real numbers. If actual infinities can exist, actual infinitesimals must exist ģ. My argument from the incoherence of actually existing infinitesimals has the following structure:Ģ.

Infinitesimals in the real number system