An alternative set theory is an alternative mathematical approach to the concept of set. It is a proposed alternative to standard set theory. In mathematics, a set can be thought of as any well-defined collection of things considered as a whole. ... Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ...

In mathematical logic, a rough set is an imprecise representation of a crisp set (conventional set) in terms of two subsets, a lower approximation and upper approximation. ... Fuzzy sets are an extension of the classical set theory used in Fuzzy logic. ...

References

Vopenka, P. Mathematics in the Alternative Set Theory. Leipzig: Teubner, 1979.

Settheory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century.

Initially controversial, settheory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties.

The most frequent objection to settheory is the constructivist view that mathematics is loosely related to computation and that naive settheory is being formalised with the addition of noncomputational elements.

Naive settheory is the original settheory developed by mathematicians at the end of the 19th century, treating sets simply as collections of things.

Axiomatic settheory is a rigorous axiomatic theory developed in response to the discovery of serious flaws (such as Russell's paradox) in naive settheory.

Internal settheory is an axiomatic extension of settheory that supports a logically consistent identification of illimited (enormously large) and infinitesimal elements within the real numbers.

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