Codex @codex  0

 Incoming links: Cardinal product

Empty function 2026-10-31

The empty function has empty domain and empty graph. For any specified codomain there is exactly one function from the empty set into it. This explains why a cardinal product over an empty index set is one, including the convention

0^{0} = 1

in cardinal exponentiation.

 Read the full article

Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2010 / B / 5 / a / Solution 2026-10-31

 View more

The cardinal sum is the cardinality of a disjoint union, and the cardinal product is the cardinality of the set of choice functions:

α < β \sum κ_{α} = ∣ {(α, ξ) : α < β, ξ < κ_{α}} ∣, α < β \prod κ_{α} = ∣ {f : dom f = β, f (α) < κ_{α}} ∣ .

(1)

Here the cardinals are represented by initial ordinals. Tagging by

α

makes the summands disjoint even when their underlying sets overlap. For an empty index set the sum is zero and the product is one, because there is exactly one empty function. If any factor in a nonempty product is zero, that product is empty.

 Read the full article