Proving Dedekind Infiniteness of Countable Sets | Solution Attempt

In summary, a set X is Dedekind infinite if there is a 1-to-1 mapping of X onto its proper subset. It can be proven that every countable set is Dedekind infinite by showing that it can be well ordered and establishing a 1-to-1 mapping to its proper subset. To be more rigorous, the mapping should be explicitly defined and the definition of countable set should be considered.
  • #1
cragar
2,552
3

Homework Statement


Call a set X Dedekind infinite if there is a 1-to-1 mapping of X onto
its proper subset.
Prove that every countable set is Dedekind infinite.

The Attempt at a Solution


I want to say that every countable set can be well ordered.
I guess I could just pick some element from our set X and call it a.
And then make sure everything from our set gets mapped to something
larger than a. So we have a 1-to-1 mapping to our proper subset.
I probably need to be more rigorous about how this mapping takes place.
 
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  • #2
cragar said:

Homework Statement


Call a set X Dedekind infinite if there is a 1-to-1 mapping of X onto
its proper subset.
Prove that every countable set is Dedekind infinite.

The Attempt at a Solution


I want to say that every countable set can be well ordered.
I guess I could just pick some element from our set X and call it a.
And then make sure everything from our set gets mapped to something
larger than a. So we have a 1-to-1 mapping to our proper subset.
I probably need to be more rigorous about how this mapping takes place.

Yes, you should be more explicit about what the mapping is. What's the definition of countable set?
 

Related to Proving Dedekind Infiniteness of Countable Sets | Solution Attempt

What is a Dedekind infinite set?

A Dedekind infinite set is a set that is infinite in the sense that it cannot be put into a one-to-one correspondence with any of its proper subsets. In other words, there is no way to pair each element in the set with a unique element in a smaller subset of the set.

Who is the mathematician associated with Dedekind infinite sets?

The concept of Dedekind infinite sets was developed by German mathematician Richard Dedekind in the late 19th century. He was one of the pioneers of set theory and his work has had a significant impact on modern mathematics.

How do Dedekind infinite sets differ from Cantor infinite sets?

Dedekind infinite sets differ from Cantor infinite sets in that they cannot be put into a one-to-one correspondence with any of their proper subsets, while Cantor infinite sets can. This means that Dedekind infinite sets are considered to be a stronger form of infinity.

What is the significance of Dedekind infinite sets in mathematics?

Dedekind infinite sets have played a crucial role in the development of modern mathematics, particularly in the field of set theory. They have also been used to prove important results in other areas of mathematics, such as calculus and topology.

Can Dedekind infinite sets be found in the real world?

While Dedekind infinite sets are an abstract concept, they can be found in the real world in certain situations. For example, the set of natural numbers (1, 2, 3, ...) is Dedekind infinite, as there is no largest natural number and it cannot be put into a one-to-one correspondence with any of its proper subsets.

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