Set theory is the branch of mathematics that studies sets, which are collections of distinct objects. Although any type of objects can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the most well known.
Set theory, formalized using first-order logic, is the most common foundational system for mathematics. The language of set theory is used in the definitions of nearly all mathematical objects, such as functions, and concepts of set theory are integrated throughout the mathematics curriculum. Elementary facts about sets and set membership can be introduced in primary school, along with Venn diagrams, to study collections of commonplace physical objects
View of the Mandelbrot set
In mathematics, the Mandelbrot set is a set of points in the complex plane, the boundary of which forms a fractal. Mathematically, the Mandelbrot set can be defined as the set of complex c-values for which the orbit of 0 under iteration of the complex quadratic polynomial xn+1=xn2 + c remains bounded. When computed and graphed on the complex plane, the Mandelbrot Set is seen to have an elaborate boundary, which does not simplify at any given magnification. This qualifies the boundary as a fractal. The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and for being a complicated structure arising from a simple definition.