analysis (differential/integral calculus, functional analysis, topology)
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continuous metric space valued function on compact metric space is uniformly continuous
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An Archimedean ordered field is an ordered field that satisfies the archimedean property.
The rational numbers are the initial ordered field, so for every ordered field there is a field homomorphism . Since every field homomorphism between ordered fields is an injection, the rational numbers is a subset of the ordered field , and we can suppress the field homomorphism via coercion, such that given one can derive that . Thus, is Archimedean if for all elements and , if , then there exists a rational number such that and .
A real number in an ordered field is an element which satisfies the Dedekind cut axioms:
We have the following results:
The first and second conditions say that every element is bounded below and above by rational numbers, and thus strictly not an infinite element. This also implies that there are no infinitesimal elements, because there are no element whose multiplicative inverse is an infinite element.
The fifth and sixth conditions independently imply that every element is strictly not an infinitesimal element.
These four conditions together imply the archimedean property for the ordered field .
The third, fourth, and seventh conditions are always true for all elements because of transitivity of the strict order relation.
Finally, the eighth condition says that every element is located, and is true for all elements because the pushout of the open intervals and with canonical inclusions and is equivalent to itself.
Thus, an Archimedean ordered field is an ordered field where every element is a real number.
Note that this definition is not the same as saying that contains every real number - the latter definition results in the Dedekind real numbers, which is the union of all Archimedean ordered fields and the terminal Archimedean ordered field.
Every Archimedean ordered field is a dense linear order. This means that the Dedekind completion of every Archimedean ordered field is the field of all real numbers.
Every element in an Archimedean ordered field satisfies the axioms of Dedekind cuts:
We have the following results:
The first condition is always true because for all , we have , and by the Archimedean principle there exists a rational number such that .
The second condition is always true because for all , we have , and by the Archimedean principle there exists a rational number such that .
The fifth condition is always true because for all and , if , then by the Archimedean principle there exists a rational number such that .
The sixth condition is always true because for all and , if , then by the Archimedean principle there exists a rational number such that .
The third, fourth, and seventh conditions are always true for all elements because of transitivity of the strict order relation.
Finally, the eighth condition says that every element is located, and is true for all elements because the union of the open intervals and is the improper subset of .
Every Archimedean ordered field is a differentiable space:
Let be an Archimedean ordered field. A function is continuous at a point if
is pointwise continuous in if it is continuous at all points :
The set of all pointwise continuous functions is defined as
Let be an Archimedean ordered field. A function is differentiable at a point if
is pointwise differentiable in if it is differentiable at all points :
The set of all pointwise differentiable functions is defined as
Let be a sub--frame of the frame of truth values. Then an Archimedean ordered field is admissible for if the pseudo-order restricts to a binary function .
The category of Archimedean ordered fields is the category whose objects are Archimedean ordered fields and whose morphisms are ring homomorphisms between Archimedean ordered fields. (Every ring homomorphism between Archimedean ordered field can be proven to be a strictly monotonic field homomorphism).
The category of Archimedean ordered fields is a thin category. It is also a skeletal category and a gaunt category, and impredicatively is the subset of the power set of real numbers which consists of all the Archimedean ordered subfields of the real numbers.
The initial object in the category of Archimedean ordered fields is the rational numbers and the terminal object in the category of Archimedean ordered fields is the (Dedekind) real numbers.
More generally, one can consider, for every sub--frame of the frame of truth values, the subcategory of Archimedean ordered fields which are admissible for and ring homomorphisms between Archimedean ordered fields admissible for . While the rational numbers are still the initial object in this category, the terminal object in this object is a version of the Dedekind real numbers, which are constructed using Dedekind cuts valued in . In the larger category of all Archimedean ordered fields, the ring homomorphism is not provably an isomorphism.
Archimedean ordered fields include
In constructive mathematics, one has the different notions of real numbers
These notions of real numbers are the same if every Dedekind real number merely has a locator, so that the Cauchy real numbers are Dedekind complete. Both excluded middle and countable choice imply that every Dedekind real number has a locator.
Non-Archimedean ordered fields include
The definition of the Archimedean property for an ordered field is given in section 4.3 of
The real numbers are defined as the terminal Archimedean ordered field and the complete Archimedean ordered field in:
Last revised on July 10, 2024 at 19:11:55. See the history of this page for a list of all contributions to it.