This picture represents infinity in terms of an infinite loop and is also a metaphor for the paradoxical qualities the infinite can possess.

Infinity, usually represented by the symbol \(\infty\) (which in words is called a "lemniscate"), is a mathematical concept that indicates a "quantity" larger than any other number. The word quantity has been placed in quotation marks to indicate that infinity goes beyond a definite quantity and in fact is not just vast, but vastless.[1] It has different meanings across different branches in mathematics.

To googologists, infinity is a cop-out from the race to invent large finite numbers. However, some mathematical infinites are useful to googologists. Ordinal infinities (transfinites) are vitally important in measuring the growth rates of functions, in particular the fast-growing hierarchy, and serve to stratify the cornucopia of computable functions that exist.

In traditional algebra, infinity is meaningful only as a symbol, not a number that can be legitimately manipulated. One use is the definition of open intervals such as \([5,\infty)\), or inequalities like \(n < \infty\). In differential and integral calculus, however, infinity is one of the central concepts. An integral, for example, is the sum of an "infinite number of infinitely small parts" — but still infinity is merely symbolic.

In complex analysis, the Riemann sphere is defined as \(\mathbb{C} \cup \{\infty\}\), where \(\infty\) is an unsigned infinity.

The term "Infinity Scraper", defined by Jonathan Bowers, refers to any number larger than tridecal. The term is, of course, hyperbolic.

Related to the "regular" mathmatical concept of infinity is complex infinity, defined as an infinite quantity which has an undefined complex argument.[2]

A super task is a task that has an infinite amount of steps but it can be completed in a finite length of time. The most famous examples of a super task are the paradoxes proposed by Greek philosopher Zeno of Elea, one of which proposes that motion is impossible as in order to traverse a distance d, you must travel half that distance, then half of that distance, and so forth infinitely. Clearly the reasoning is at fault with our common experience, but it goes to show just some of the contradictions that arise when super tasks are invoked.

Sources Edit

  1. 1.1.1 Welcome to the Numberscape - Large Numbers
  2. Complex Infinity - from Wolfram MathWorld