Applications:

The concept of infinity is used widely throughout mathematics. The idea of letting things ''go to infinity'' or ''approach infinity'' is where the idea of limits came from. The development of calculus could not have occurred without the infinite. Both trigonometry and geometry often include definitions relying on our idea of infinite, such as the definition of a circle as a polygon with an infinite number of sides. Set theory and Cantor's work allowed for strides in various fields of mathematics: number theory (the study of numbers, especially whole numbers and rational numbers); analysis (which includes differentiation, integration, limits, and infinite series); and topology (''rubber sheet geometry,'' or the study of shapes and spaces).

There are also applications outside the realm of mathematics, especially in the fields of metaphysics, theology and philosophy. However, there are problems when we try to move from more abstract mathematics into real-life.

Controversies:

Infinity causes problems in physics. For example, when physicists first worked on the the quantum theory of electromagnetic force, they initially found the mass and charge of an electron to be infinite. Or when considering the Big Bang Theory, treating space-time as infinite would imply that inflation would never stop, and that ''big bangs'' would continue to occur an infinite number of times.

Some are again trying to redefine mathematics so as to avoid infinity. Norman Wildberger has been working on developing an infinity free trigonometry and Euclidean geometry by avoiding the irrational (and infinite) number pi to define angles and instead turning to rational output from vectors. Doron Zeilberger is trying to get rid of not just actual infinity, but also potential infinity, by claiming that there is indeed a largest number (which he calls N0).

One way of dealing with the unstable nature of infinity in traditional mathematics lies in the use of hyperreals, R*. Abraham Robinson developed the hyperreals in 1960, and it includes all ''standard'' numbers in R and also the ''nonstandard'' (infinitely large or small) numbers. This alternative to Cantorian set theory provides a rigorous and consistent mathematical foundation. For example, a standard definition of limits would read: ''Let f be a function defined on some open interval (a,b). The limit of f(x) as x approaches a is equal to the value L, if for every e>0 there exists d>0 such that |f(x)-L| is less than e whenever x-a is between 0 and d.'' A non-standard definition would be: ''f(x)-L is infinitesimal whenever x-a is infinitesimal.'' The advantage to using the hyperreals is that all the normal operations under R are preserved.