What Is Infinity Plus One?
The expression ∞+1 has no meaning in ordinary real arithmetic, because infinity is not a real number. It acquires a meaning only after the mathematical context has been specified.
The most familiar context is calculus, where +∞ describes unbounded behavior. Analysis can also enlarge the real line by adjoining two infinite elements. Set theory introduces a different distinction: cardinal numbers measure the size of sets, whereas ordinal numbers describe ordered arrangements. The same informal phrase, infinity plus one, therefore corresponds to several different statements.
Infinity in Limits
Consider
\[\lim_{x\to+\infty}(x+1)=+\infty.\]This does not result from substituting an infinite number for x. It states that x+1 eventually exceeds every prescribed real bound. More precisely, for every real number M, there exists a real number N such that
\[x>N \quad\Longrightarrow\quad x+1>M.\]Adding one does not alter the unbounded behavior of the function. The same is true more generally when a fixed real constant is added:
\[f(x)\to+\infty \quad\Longrightarrow\quad f(x)+c\to+\infty.\]The statement concerns the eventual values of a function, rather than an arithmetic operation performed on an object called infinity. This is also why expressions such as
\[+\infty-\infty\]cannot be simplified algebraically in a limit. Two functions may both tend to +∞, while their difference tends to a finite number, tends to either infinity, or fails to have a limit.
The Extended Real Line
For some purposes, analysis enlarges the real line by adjoining two elements:
\[\overline{\mathbb R} = \mathbb R\cup\{-\infty,+\infty\}.\]In this system one defines, for every real number a,
\[+\infty+a=+\infty, \qquad -\infty+a=-\infty.\]Consequently,
\[+\infty+1=+\infty\]is a valid formula in the extended real line. It expresses an arithmetic convention compatible with the order of the extension: +∞ remains greater than every real number after a finite quantity is added.
The extended real line is not a field. Operations such as
\[+\infty+(-\infty), \qquad 0\cdot\infty, \qquad \frac{\infty}{\infty}\]remain undefined. The familiar algebraic rules of the real numbers therefore cannot be transferred to it without restrictions.
Cardinal Numbers
Cardinal numbers measure the size of sets. Two sets have the same cardinality when there is a bijection between them: every element of the first set is paired with exactly one element of the second, and conversely.
The natural numbers have cardinality
\[|\mathbb N|=\aleph_0.\]To add two cardinal numbers, one takes two disjoint sets having those cardinalities and measures the size of their union. If
\[|A|=\kappa, \qquad |B|=\lambda, \qquad A\cap B=\varnothing,\]then cardinal addition is defined by
\[\kappa+\lambda=|A\cup B|.\]The requirement that the sets be disjoint prevents a common element from being counted only once. When the original sets overlap, disjoint copies can be formed by attaching different labels:
\[A\times\{0\}, \qquad B\times\{1\}.\]The labels serve only to distinguish the two copies.
Now add one new element, denoted by a, to the natural numbers. The cardinal sum ℵ₀+1 is the size of
\[\mathbb N\cup\{a\},\]where a∉ℕ. Define a map from this set to ℕ by
\[f(a)=0, \qquad f(n)=n+1 \quad\text{for }n\in\mathbb N.\]Every element of ℕ∪{a} receives a different natural number, and every natural number is reached. The map is therefore a bijection, so
\[\aleph_0+1=\aleph_0.\]The equality compares sizes. It does not assert that the new element disappears; it asserts that the enlarged set can still be placed in one-to-one correspondence with the natural numbers.
The same method gives
\[\aleph_0+\aleph_0=\aleph_0.\]Take two labelled copies of ℕ. Map the first copy to the even numbers and the second to the odd numbers:
\[(n,0)\longmapsto 2n, \qquad (n,1)\longmapsto 2n+1.\]Together the two copies still form a countable set. Likewise, adding any finite number m of elements gives
\[\aleph_0+m=\aleph_0.\]Hilbert’s Hotel is a spatial version of the first bijection. If every room numbered by a natural number is occupied, moving the guest in room n to room n+1 leaves room 0 free. The rearrangement illustrates why adjoining one element does not change countable cardinality; the bijection is the mathematical argument.
Ordinal Numbers
Cardinality ignores order. Ordinal numbers retain it.
The ordinal
\[\omega\]is the order type of the natural numbers in their usual order:
\[0<1<2<3<\cdots.\]Ordinal addition is defined by placing one ordered set after another. Thus α+β is obtained by taking an ordered copy of α and placing a copy of β after all its elements.
If one new element is placed after the entire sequence of natural numbers, the resulting order has type
\[\omega+1.\]This order has a greatest element, whereas ω has none. They cannot be order-isomorphic, and therefore
\[\omega+1\ne\omega.\]They do, however, have the same cardinality:
\[|\omega+1|=|\omega|=\aleph_0.\]Placing the new element before the natural numbers gives
\[1+\omega.\]The order
\[a<0<1<2<3<\cdots\]is order-isomorphic to the natural numbers: send a to 0, the old 0 to 1, the old 1 to 2, and so on. Hence
\[1+\omega=\omega, \qquad \omega+1>\omega.\]Ordinal addition is therefore not commutative. The position of the added element changes the order type even when it does not change the number of elements.
The Four Meanings
The original expression can now be separated into four claims:
- In ordinary real arithmetic, ∞+1 is not defined.
- In a limit, f(x)+1→+∞ describes unbounded behavior.
- In the extended real line, +∞+1=+∞ is one of the rules of the extension.
- In set theory, ℵ₀+1=ℵ₀ for cardinal addition, whereas ω+1≠ω for ordinal addition.
The notation alone does not determine which of these claims is intended. The relevant mathematical structure must be specified before the expression can be evaluated.