nLab product-regular cardinal

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Contents

Context

Universes

Foundations

foundations

The basis of it all

 Set theory

set theory

Foundational axioms

foundational axioms

Removing axioms

Contents

Idea

The version of regular cardinal for products/Cartesian products instead of sums/unions/disjoint unions.

Definition

A cardinal κ\kappa is product-regular if, for all families of sets (X i) iI(X_i)_{i \in I}, if |I|<κ\vert I \vert \lt \kappa and for all elements iIi \in I |X i|<κ\vert X_i \vert \lt \kappa, then the indexed product

| iIX i|<κ\left| \prod_{i \in I} X_i \right| \lt \kappa

Properties

Every inaccessible cardinal is product-regular. Every uncountable product-regular cardinal is an inaccessible cardinal, but the finite cardinals 00 representing the empty set and the countable cardinal 0\aleph_0 representing the natural numbers are product-regular cardinals which are not inaccessible cardinals.

See also

References

Created on September 28, 2022 at 16:04:47. See the history of this page for a list of all contributions to it.