##  [Ultrapower](/ultrapower) 

  ##  [Ultrapower](https://mathlogic.quantumdictionary.io/ultrapower-0) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A structure obtained from a given structure M by taking the Cartesian product M^I of copies of M indexed by a set I and then factoring by an ultrafilter U on I: elements are equivalence classes of functions I → M that agree on a set in U. Ultrapowers and more general ultraproducts are central tools for producing elementary extensions and nonstandard models.

 

 

 

 

 





 

 



 ##  [Ultrapower](https://puremath.quantumdictionary.io/ultrapower-1) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

An ultrapower of a structure M by an ultrafilter U on an index set I is the ultraproduct of the constant family (M)_{i∈I} modulo U; elements are equivalence classes of sequences in M, and Łoś's theorem makes the ultrapower an elementary extension of M under nonprincipal ultrafilters.