Understanding the Order Types of n + ω and ω + n

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In summary, the conversation discusses the difference between the order types of n + ω and ω + n, and how they are related to the set of natural numbers and the set of natural numbers with additional elements. It is explained that n + ω is order-isomorphic to a set where n elements are added after the natural numbers, while ω + n is not isomorphic to this set. This is because elements in ω + n have an infinite number of predecessors, whereas elements in ω have a finite number of predecessors.
  • #1
Glinka
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Hi all,

Went over this today and I'm not grasping it: why is the order type of n + ω = ω, while ω + n ≠ ω? I'd really appreciate if someone could set up the requisite isomorphism in the former. Thanks!
 
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  • #2
The set [itex]n+\omega[/itex] is essentially (order-isomorphic to) the following:

[tex](0,0)<(1,0)<(2,0)<(3,0)<...<(n-1,0)<(0,1)<(1,1)<(2,1)<(3,1)<...<(k,1)<...[/tex]

Do you see that??

The isomorphism between the above set and [itex]\omega[/itex] is given by the map T that does the following:

[tex]T(k,0)=k,~T(k,1)=n+k[/tex]
 
  • #3
Ahh yes, this helps a lot, thanks. So in case of $$\omega + n $$ we could try $$(0,0) < (1,0) < ... < (k,0) < ... < (0,1) < (1,1) < ... < (n-1,1)$$ but we wouldn't be able to set up an isomorphism between this and ##\omega##?
 
  • #4
Glinka said:
Ahh yes, this helps a lot, thanks. So in case of $$\omega + n $$ we could try $$(0,0) < (1,0) < ... < (k,0) < ... < (0,1) < (1,1) < ... < (n-1,1)$$ but we wouldn't be able to set up an isomorphism between this and ##\omega##?

Yeah exactly. Here we have the natural numbers and we paste n elements after it.
So take (0,1) for example. That has an infinite number of predecessors. So it can't be [itex]\omega[/itex] since any element in [itex]\omega[/itex] has a finite number of predecessors.
 
  • #5
Excellent, thanks for your help!
 

FAQ: Understanding the Order Types of n + ω and ω + n

What is the difference between n + ω and ω + n order types?

The main difference between these two order types is the way they are constructed. In n + ω, the first element is a finite number (n) followed by an infinite number (ω). In ω + n, the first element is an infinite number (ω) followed by a finite number (n).

How are these order types related to the concept of infinity?

Both n + ω and ω + n order types involve infinite numbers, but in different positions. This reflects the different ways in which infinity can be approached, whether it is added to a finite number or preceded by a finite number.

Can these order types be visualized?

Yes, these order types can be visualized using a number line. The number line would start with the finite number n and continue towards infinity (represented by ω). In the case of ω + n, the number line would start with ω and extend towards the finite number n.

How do these order types differ from other infinite order types?

While n + ω and ω + n involve infinite numbers, they are different from other infinite order types such as ω, ω², and ωω. These other types involve the repetition of a single number or pattern, while n + ω and ω + n involve a finite number followed by an infinite number.

What are some real-world applications of these order types?

These order types have applications in set theory, topology, and other branches of mathematics. They can also be used to study the behavior of functions and sequences in calculus and analysis. Additionally, they have applications in computer science, particularly in the study of algorithms and data structures.

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