Large Complete Ordered Fields

In summary, there is a theorem that states that every ordered field with the least upper bound property is isomorphic to the real numbers. This means that any ordered field with the least upper bound property must have the same cardinality as the real numbers, which is c. This applies to ordered fields that are order-complete, meaning they have the least upper bound property. Since complete fields have to contain limits of all sequences of its elements, there cannot be any other elements aside from those described, which have a cardinality of 2^\aleph_0. This means that there are no complete ordered fields with a cardinality greater than 2^{\aleph_0}.
  • #1
Dragonfall
1,030
4
I was told that there are no complete ordered fields of cardinality greater than [tex]2^{\aleph_0}[/tex]. Why is that?
 
Physics news on Phys.org
  • #2
i'm not 100% sure about this but..

I remember hearing a theorem that says that every ordered field with the least upper bound property is isomorphic to the reals. So every ordered field that is order-complete (ie has least upper bound property) has to have cardinality c.

hmmm..i'm not sure order-complete and complete are always equivalent
 
  • #3
Let's think about it:

A field has to have 0 and 1, and hence all things of the form a/b for a,b sums of 1. This has cardinality at most |Q| the rationals, i.e. aleph_0. Now it has to be complete, so it must contain limits of all sequences of these elements. There are 2^\aleph_0 of these. The only question is now if there can be any other element. No there can't - if there were some element x I've not described, then I can assume it is positive by looking at -x if necessary, and if it's larger than 1, I can replace it with 1/x. Thus I have to show that any x such that 0<x<1 has already been described. But this is true, since I can construct a sequence of 'rationals' that approximate it by repeated bisection of the interval.
 

1. What is a large complete ordered field?

A large complete ordered field is a mathematical structure that combines the properties of a field (a set with two binary operations, addition and multiplication) and an ordered set (a set with a relation that is transitive, reflexive, and total) in a way that satisfies certain additional conditions. In simpler terms, it is a set of numbers that can be added, subtracted, multiplied, and divided, and also has a defined order among its elements.

2. How does a large complete ordered field differ from a regular field?

A large complete ordered field differs from a regular field in that it has an additional property of completeness. This means that every non-empty subset of the field that is bounded above (or below) has a least upper bound (or greatest lower bound) within the field. In other words, there are no "gaps" in the set of numbers, and every sequence of numbers that approaches a certain limit will have that limit within the field.

3. What is the significance of completeness in a large complete ordered field?

The completeness property in a large complete ordered field is significant because it allows for the existence of solutions to certain mathematical problems that may not have a solution in a regular field. For example, the completeness of the real numbers (a specific type of large complete ordered field) allows for the existence of solutions to equations such as x2 = 2, which has no solution in the rational numbers (a regular field).

4. Can large complete ordered fields be used in practical applications?

Yes, large complete ordered fields have practical applications in various fields, including economics, physics, and computer science. In economics, they are used to model and analyze decision-making processes with uncertainty. In physics, they are used to represent and solve problems involving continuous quantities, such as time and space. In computer science, they are used in algorithms and data structures for efficient searching and sorting of data.

5. Are there any other properties or characteristics of large complete ordered fields to be aware of?

Yes, there are other important properties and characteristics of large complete ordered fields, such as the Archimedean property (which states that there is no infinite element in the field) and the density property (which states that between any two distinct elements, there exists another element in the field). These properties are essential in understanding the behavior and applications of large complete ordered fields.

Similar threads

  • Calculus
Replies
2
Views
1K
Replies
2
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
7
Views
1K
Replies
1
Views
830
  • Set Theory, Logic, Probability, Statistics
Replies
16
Views
1K
  • Classical Physics
Replies
13
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
539
  • Electromagnetism
Replies
8
Views
754
Replies
3
Views
2K
Back
Top