# How can infinitesimals be invertible in SIA?

I read that infinitesimals in SIA can be invertible: https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis

In typical models of smooth infinitesimal analysis, the infinitesimals are not invertible, and therefore the theory does not contain infinite numbers. However, there are also models that include invertible infinitesimals.

How is it possible if SIA use intuitionistic logic?

Thanks.

• The nilsquare/nilpotent infinitesimals of SIA are never invertible. Every model of SIA has nilpotent infinitesimals, and normally, when we refer to "infinitesimals" in SIA, we mean these nilpotent infinitesimals. However, some (but not all) models of SIA also include other, non-nilpotent infinitesimals alongside the nilpotent ones. This other type of infinitesimal works very similarly to the infinitesimals from nonstandard analysis: smaller than every positive rational number, but still provably nonzero, and therefore invertible. Commented 2 days ago
• @Z.A.K. Big thanks! Commented 2 days ago

Models of SIA can include infinite numbers $$H$$. When this happens, $$\frac1{H}$$ is an infinitesimal which is provably nonzero and therefore different from the nilsquare infinitesimals.