Infinity …    
      … it’s not big …
      … it’s not huge …
      … it’s not tremendously large …
      … it’s not extremely humongously enormous …
      … it’s


Infinity has no end. Infinity is the idea of something that has no end.

"Paul Erdős lived in Budapest, Hungary, with his Mama. Mama loved Paul to infinity ∞. When Paul was 3. She had to go back to work as a math teacher….” (Extract from the book: The Boy Who Loved Math: The Improbable Life of Paul Erdős by Deborah Heiligman - Figure 1).

Infinity, most often denoted as infty(symbol:∞), is an unbounded quantity that is greater than every real number, is an abstract concept describing something without any limit and is relevant in a number of fields, predominantly mathematics and physics. In number systems incorporating infinitesimals, the reciprocal of an infinitesimal is an infinite number, i.e., a number greater than any real number. Infinity is a very tricky concept to work with, as evidenced by some of the counterintuitive results that follow from Georg Cantor’s treatment of infinite sets.
Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities). For example, the set of integers is countably infinite, while the infinite set of real numbers is uncountable. (Here is one of Proofs)

  • In Geometry and topology: Main article: Dimension (vector space). Infinite-dimensional spaces are widely used in geometry and topology, particularly as classifying spaces, notably Eilenberg−MacLane spaces. Common examples are the infinite-dimensional complex projective space K(Z,2) and the infinite-dimensional real projective space K(Z/2Z,1).
  • In Fractal Geometry: The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming “smooth”; they have infinite perimeters—some with infinite, and others with finite surface areas. One such fractal curve with an infinite perimeter and finite surface area is the Koch snowflake.
  • In Real analysis: In real analysis, the symbol \infty, called “infinity”, is used to denote an unbounded limit. x -> ∞ means that x grows without bound, and x  -> - ∞ means the value of x is decreasing without bound.

See more at: Infinity on Wikipedia and Mathworld - What is Infinity? on MathisFun.

Reference:  Paul Erdös and the Erdös Number Project page.

Image: Koch snowflakes & The Boy Who Loved Math: The Improbable Life of Paul Erdős.