Such systems include:
- Abstract Stone duality, a programme to re-axiomatise general topology directly, instead of using set theory. It is formulated in the style of type theory and is in principle computable. It is currently able to characterise the category of (not necessarily Hausdorff) computably based locally compact spaces. It allows the development of a form of constructive real analysis using topological rather than metrical arguments.
- Chainlet geometry, a recent development of geometric integration theory which incorporates infinitesimals and allows the resulting calculus to be applied to continuous domains without local Euclidean structure as well as discrete domains.
- Constructive analysis, which is built upon a foundation of constructive, rather than classical, logic and set theory.
- Intuitionistic analysis, which is developed from constructive logic like constructive analysis but also incorporates choice sequences.
- p-adic analysis.
- Paraconsistent analysis, which is built upon a foundation of paraconsistent, rather than classical, logic and set theory.
- Smooth infinitesimal analysis, which is developed in a smooth topos.
Non-standard analysis and the calculus it involves, non-standard calculus, are considered part of classical mathematics (i.e. The concept of "hyperreal number" it uses, can be constructed in the framework of Zermelo–Fraenkel set theory).
- "Paul Taylor's site". Paultaylor.eu. Retrieved 2013-09-23.