Infinity plus one

In mathematics, infinity plus one has meaning for the hyperreals, and also as the number ω+1 (omega plus one) in the ordinal numbers and surreal numbers. This phrase is also found in popular culture.

Mathematics

There are several mathematical theories which include both infinite values and addition.

• Cardinal numbers are representations of sizes (cardinalities) of abstract sets, which may be infinite. Addition of cardinal numbers is defined as the cardinality of the disjoint union of sets of given cardinalities. It can be easily shown that κ + 1 = κ for any infinite cardinal κ, as illustrated by Hilbert's paradox of the Grand Hotel. Moreover, if one assumes the axiom of choice, then κ + λ = max { κ, λ } if at least one of κ or λ is infinite.
• Ordinal numbers represent order types of well-ordered sets. Ordinal addition is defined as the order type of the concatenation of orders. This operation is not commutative: ω + 1 is a strictly larger ordinal than ω, but 1 + ω = ω.
• Hyperreal numbers are an extension of the real number system which contains infinite and infinitesimal numbers. The resulting system is an ordered field thanks to the transfer principle, which states that any first-order sentence which is true for real numbers also holds for hyperreals. Since ∀x: x < x + 1 is a first-order sentence holding for reals (as it follows from the ordered field axioms), adding one to an infinite hyperreal produces a larger value. The same will hold for any non-Archimedean ordered field.
• Surreal numbers also extend real numbers to a system which satisfies the axioms of an ordered field, and so addition behaves similarly to hyperreals, in that x < x + 1 for all surreals x. In this system, one may find elements corresponding to infinite ordinals; however, surreal addition and multiplication correspond not to the usual ordinal operations, but to the natural sum and natural product.

Popular culture

Infinity plus one is a phrase used in relation to the notion of infinity as the largest possible amount. The idea is that no such number should exist.

The phrase is also 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,^[1] 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, My Little Pony: Friendship Is Magic, Timon and Pumbaa.

One "Dilbert" strip showed an enemy of the title character promising Bob the Dinosaur "infinity plus two" times the wages Dilbert paid him.

See also

• Beyond the Infinite
• Eternity and a Day

Notes

References