Homotopy Type Theory
generalized Cauchy real numbers > history (Rev #3, changes)
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Definition
Let be a dense integral subdomain of the rational numbers and let be the positive terms of .
In set theory
The Letgeneralized Cauchy real numbers be therational numbers are and inductively let generated by the following:
-
a function , where
is the type of all Cauchy nets with index type and values in .
-
a dependent family of terms
where and are directed types and the equivalence relation on Cauchy nets is defined as
-
a term
be the set of positive rational numbers. Let be a directed set and
is the set of all Cauchy nets with index set and values in . Let
be the (large) set of all Cauchy nets in in a universe , where is the category of all directed sets in .
Let the relation in the Cartesian product for directed sets and be defined as
The generalized Cauchy real numbers is a set with a function such that
In homotopy type theory
Let be the rational numbers and let
be the positive rational numbers.
The generalized Cauchy real numbers are inductively generated by the following:
-
a function , where
is the type of all Cauchy nets with index type and values in .
-
a dependent family of terms
where and are directed types and the equivalence relation on Cauchy nets is defined as
-
a term
See also
Revision on April 14, 2022 at 01:28:52 by
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