Infinity plus one

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 infinity. 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.