Archimedean property for unbounded sets

In summary, there is a proof of the Archimedean property in the book that relies on the existence of sups, which in turn relies on the existence of a bound. However, it is unclear if this property holds true for unbounded sets, specifically the real numbers, and whether the OP is asking about sets that do not satisfy the Axiom of Completeness. The rational numbers serve as an example of a set that satisfies the Archimedean property without satisfying the Axiom of Completeness. Further clarification and potentially citing the text in question would be helpful.
  • #1
Does the Archimedean property work for unbounded sets? My book does a proof of the Archimedean property relying on the existence of sup which relies on the existence of a bound.
 
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  • #2
torquerotates said:
Does the Archimedean property work for unbounded sets? My book does a proof of the Archimedean property relying on the existence of sup which relies on the existence of a bound.

Unbounded sets of what type of elements? Real numbers? I think you should reword or provide more details of your question and maybe even cite the text you're referring to (including page number, theorem number, etc.).
 
  • #3
The real numbers are unbounded and have the Archimedean property.
 
  • #4
If I'm recalling the proof of the Archimedean property correctly, I wonder if the OP actually meant to ask if this property holds true for sets that do not satisfy the Axiom of Completeness (i.e., the existence of sups)...

I suppose the rational numbers are an example of a set that satisfies the Archimedean property without satisfying the Axiom of Completeness.
 

1. What is the Archimedean property for unbounded sets?

The Archimedean property for unbounded sets states that for any two unbounded sets, there exists a natural number n such that the first set contains at least n elements more than the second set.

2. How is the Archimedean property for unbounded sets different from the Archimedean property for bounded sets?

The Archimedean property for bounded sets states that for any two bounded sets, there exists a natural number n such that the first set contains at least n elements more than the second set. The difference is that for unbounded sets, there is no limit to the number of elements in each set, while for bounded sets, there is a specific upper limit.

3. Why is the Archimedean property important in mathematics?

The Archimedean property is important because it allows us to compare the sizes of different sets. This is useful in many areas of mathematics, such as measure theory and calculus, where we need to determine the relative sizes of sets.

4. Can the Archimedean property be extended to infinite sets?

Yes, the Archimedean property can be extended to infinite sets. This is known as the Archimedean principle, which states that for any two infinite sets, there exists a bijection between them. This means that the two sets have the same size, even though they may contain an infinite number of elements.

5. Are there any real-life applications of the Archimedean property for unbounded sets?

Yes, the Archimedean property for unbounded sets is used in the field of economics to compare the sizes of infinite sets, such as the number of goods and the number of consumers in a market. It is also used in probability theory to calculate the likelihood of events in infinite sample spaces.

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