Where does cardinal collide?

In set theory, a cardinal number measures the size of a set (how many elements it contains), generalised by Cantor so that even infinite sets can be compared: two sets have the same cardinality exactly when a one-to-one correspondence (bijection) exists between them, which is how the infinities aleph-null and the continuum are ranked. Halmos's Naive set theory develops cardinals in this way. A cardinal answers “how many,” and is distinct from an ordinal, which encodes order or position; it has nothing to do with lens reference points or clinical signs.[1]

Unit

The cardinality: a measure of set size, finite or transfinite

Recovery means

Not applicable

Measured by

Established by exhibiting a bijection (equinumerosity) between sets

In optics, the cardinal points of an imaging system are the special reference points on the optical axis (the two focal points, the two principal points, and the two nodal points) that together fully capture the first-order (Gaussian) imaging behaviour of any lens or mirror assembly, so that a thick lens or a complex system can be treated as a simple equivalent. They are set out in Born and Wolf's Principles of optics. The referent is a set of locations fixing paraxial image formation, and “cardinal” here means principal or pivotal, not a kind of number or a symptom.[2]

Unit

The axial positions of the focal, principal, and nodal points

Recovery means

Not applicable

Measured by

Located by paraxial ray tracing and Gaussian optics of the system