Abstract:

The nonstandard approach to fuzzy sets is based on a Boolean generalization of Infinitesimal Analysis. This paper, gives a short review of this approach, describes some applications to mathematical structures and indicates the way for an extension using fuzzy partitions. In addition, we prove that the theory is general, since for any ordinary fuzzy set $f: X\to [0,1]$ there exists a unique Boolean probability algebra $(\B,p)$ and a $\B$-possibility distribution $\pi :X\to\B$, such that $$ f=p\circ \pi . $$