Infinity plus one

In popular culture, infinity plus 1 is a phrase used in relation to the notion of infinity as the largest possible number. The idea is that no such number should exist.

In mathematics, however, "infinity plus 1" has meaning as the number ω + 1 in the ordinal numbers and surreal numbers. Alternatively, if "infinity" is interpreted as any specific cardinal number κ ≥ $aleph\_0$ in cardinal arithmetic, then "infinity plus 1" = κ + 1 = κ.

The phrase is sometimes found in children's culture by those who have discovered the concept of infinity. When one child makes a reference to infinity, another may try to best them by making a similar reference to "infinity plus one". One example of this is when children come up with a dare, and subsequently return the dare, multiplied by increasingly higher values, for example:

:"Child 1: "I dare you ...":"Child 2: "I double dare you.":"Child 1: "I triple dare you.":"Child 2: "I dare you times a million.":"Child 1: "I dare you times infinity.":"Child 2: "I dare you times infinity plus one."

This sort of exchange has been used in many popular television programs, such as The Simpsons and Timon and Pumbaa.

References

*cite journal |last=Monaghan |first=John |title=Young Peoples' Ideas of Infinity |format= |journal=Educational Studies in Mathematics |year=2001 |volume=48 |pages=239–257 |doi=10.1023/A:1016090925967
*cite journal |last=Shulman |first=Polly |title=Infinity Plus One, and Other Surreal Numbers |journal=Discover |volume=16 |issue=12 |year=1995 |month=December |url=http://www.discover.com/issues/dec-95/features/infinityplusonea599/ |format=dead link|date=June 2008 – ^{[http://scholar.google.co.uk/scholar?hl=en&lr=&q=author%3AShulman+intitle%3AInfinity+Plus+One%2C+and+Other+Surreal+Numbers&as_publication=Discover&as_ylo=1995&as_yhi=1995&btnG=Search Scholar search]}
*cite journal |last=Tall |first=David |title=A child thinking about infinity |journal=Journal of Mathematical Behavior |volume=20 |issue=1 |year=2001 |pages=7–19 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001l-childs-infinity.pdf |doi=10.1016/S0732-3123(01)00058-X