Real Numbers: Axioms or Theorem?

It seems that the set of real numbers under the usual sum and product is considered a field because it satisfies the field axioms, but it is proved as a theorem rather than assumed as an axiom.
  • #1
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Hi there,

In most books that I saw, the set of real numbers under the usual sum and product is considered as a Field and say that's by the field axioms. But I have surprised when I have seen, it is a theorem. The question, are these axioms? or can they be proved?

Thank you very much.
 
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  • #2
What are "axioms" and what are "theorems" depends upon what level you are working at.

We define a field to be a set of objects, S, together with two binary operations, + and *, satisfying the "field axioms".

We can the prove, as a theorem, that a particular set, with given binary operations, such as the rational numbers or real numbers, satisfy those axioms.
 
  • #3
Thank you
 

1. What are real numbers?

Real numbers are a set of numbers that include all rational and irrational numbers. These numbers can be represented on a number line and can be used to measure quantities such as length, time, and temperature.

2. What are the axioms of real numbers?

The axioms, or basic properties, of real numbers include closure, commutativity, associativity, distributivity, identity, and inverses. These properties allow for operations such as addition, subtraction, multiplication, and division to be performed on real numbers.

3. Why are real numbers important in mathematics?

Real numbers are important in mathematics because they are used to model and solve real-world problems. They also form the foundation for other branches of mathematics, such as calculus and linear algebra.

4. How are real numbers different from other types of numbers?

Real numbers are different from other types of numbers, such as complex numbers or imaginary numbers, because they can be represented on a number line and have a specific order. They also include both rational and irrational numbers, while other types of numbers may only include one or the other.

5. What is the difference between an axiom and a theorem in regards to real numbers?

An axiom is a fundamental property or assumption that is used to define and prove other properties or theorems. A theorem, on the other hand, is a statement that has been proven using axioms and other theorems. In the context of real numbers, axioms are used to define the properties of real numbers, while theorems are used to prove specific statements about real numbers.

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