- #1

- 207

- 0

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

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

- 207

- 0

Physics news on Phys.org

- #2

- 5

- 0

torquerotates said:

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

- 147

- 0

The real numbers *are* unbounded and have the Archimedean property.

- #4

- 360

- 0

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

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.

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.

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.

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.

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.

Share:

- Replies
- 10

- Views
- 1K

- Replies
- 18

- Views
- 2K

- Replies
- 2

- Views
- 982

- Replies
- 12

- Views
- 1K

- Replies
- 2

- Views
- 1K

- Replies
- 1

- Views
- 787

- Replies
- 16

- Views
- 3K

- Replies
- 2

- Views
- 843

- Replies
- 9

- Views
- 1K

- Replies
- 23

- Views
- 2K