lol

rofl

N_^O E_^XCEPTIONS

Cardinality is studied for its own sake as part of set theory. It is also a tool used in branches of mathematics including combinatorics, abstract algebra, and mathematical analysis. In category theory, the cardinal numbers form a skeleton of the category of sets.

>>3
what is cardinality good for?

All the remaining propositions in this section assume the axiom of choice:

    If κ and μ are both finite and greater than 1, and ν is infinite, then κ^ν = μ^ν.

    If κ is infinite and μ is finite and non-zero, then κ^μ = κ.

>>4
Oppressing the goyim.

The practice of refusing infinite values for measurable quantities does not come from a priori or ideological motivations, but rather from more methodological and pragmatic motivations.

Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using Venn diagrams. The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition.

The surreal numbers are a proper class of objects that have the properties of a field.

Hence S breaks up into uncountably many orbits under G. Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset X of S with the property that all of its translates by G are disjoint from X. The set of those translates partitions the circle into a countable collection of disjoint sets, which are all pairwise congruent.

For example, if one defines categories in terms of sets, that is, as sets of objects and morphisms (usually called a small category), or even locally small categories, whose hom-objects are sets, then there is no category of all sets, and so it is difficult for a category-theoretic formulation to apply to all sets.