Confusion with finite sets vs an infinite set of natural numbers

[Boltzmannbrain]

17 minutes ago, wtf said:

You've discovered the ordinal numbers! The first transfinite ordinal is called ω , the lower-case Greek letter omega. It's a number that "comes after" all the finite natural numbers.

In set theory it's exactly the same set as ℵ0 but considered as an ordinal (representing order) rather than a cardinal (representing quantity).

So the ordinal number line begins:

0, 1, 2, 3, 4, ... ω , ...

Now the point is, there is no "last" natural number n that "reaches" or "is right before" ω . It doesn't work that way. If you are at ω and you take a step backwards, you will land on some finite natural number. But there are still infinitely many other natural numbers to the right of the one you landed on.

You can jump back from ω to some finite natural number (which still has infinitely many natural numbers after it), but you can't jump forward a single step to get back to ω .

That's just how it works. 

There's even a technical condition that lets us recognize why ω is special.

A successor ordinal is an ordinal that has an immediate predecessor. All the finite natural numbers except 0 are successor ordinals.

A limit ordinal is an ordinal that has no immediate predecessor. ω is a limit ordinal. That is, there is no other ordinal whose successor is ω .

Note also that by this definition, 0 is also a limit ordinal. It's the only finite limit ordinal.

 

 

Okay, I have read some about ordinals.  What about my lines below?  How do the ordinals come into play?  

 

I want to map every n from the set of natural numbers inclusively between the points shown on the finite line below from the origin to the X.  I want to do this by us imagining that there are points on the line of every rational number between the origin and X. 

 

Can the point at X be one of the rationals that I want n to list to? 

 

                                                   Origin ._______________________________________________. X  

 

There is an aleph null infinity of rational numbers inclusively between the two points of the origin and the X.  Since that I am mapping an n of N to all rational numbers, wouldn't there have to be an natural number at the point of the X?  

 

Another contradictory result is that if we zoom in to these enumerated rational points on the line, it seems to give us a first rational segment which I don't think is allowed.

 

Origen .        1.        2.        3.        4.        .        .        .   (aleph null dots) . X

 

Can't I use only natural numbers instead of ordinals? 

[wtf]

14 minutes ago, Boltzmannbrain said:

Can't I use only natural numbers instead of ordinals? 

You can bijectively map the natural numbers to your Boltzmann line (B-line) as follows:

0 <-> X

1 <-> 0

2 <-> 1

3 <-> 2

...

It's perfectly clear that you can do that, since your B-line and the naturals have the same cardinality.

What you can NOT do is map them bijectively in an order-preserving manner. Why is that? Because they have a different order type. The natural numbers have no largest element, while the B-line does have a largest element, namely X.

So: There is a bijection, but not an order-preserving bijection, between the points of the B-line and the points of the natural numbers.

The two sets are cardinally equivalent, but not ordinally equivalent.

Edited May 5, 2023 by wtf

[joigus]

On 5/4/2023 at 5:32 PM, Boltzmannbrain said:

I agree, but I don't see how this helps solve my issue.

Your issue is a non-issue. You don't make any sense. You never do, nor do you seem to care. Having an infinite element (in a particular sense that in the case of natural numbers is clear, and identifiable with a norm) or having infinitely many elements in a set (cardinality), or having a measure of a set are different things.

You are --deliberately or not, I don't know-- confusing whether an element is finite (norm?) with how many elements there are in a set (cardinal?), or perhaps a measure (some concept of "extension" or "volume"). One way or another, several members are trying to help you grope towards these important concepts in mathematics, but you don't seem to care, and keep demanding them to address your silly "analogies."

BTW, @wtf's last comments go in the direction of your pretence confusion.

 

 

[Boltzmannbrain]

35 minutes ago, wtf said:

You can bijectively map the natural numbers to your Boltzmann line (B-line) as follows:

0 <-> X

1 <-> 0

2 <-> 1

3 <-> 2

...

It's perfectly clear that you can do that, since your B-line and the naturals have the same cardinality.

What you can NOT do is map them bijectively in an order-preserving manner. Why is that? Because they have a different order type. The natural numbers have no largest element, while the B-line does have a largest element, namely X.

So: There is a bijection, but not an order-preserving bijection, between the points of the B-line and the points of the natural numbers.

The two sets are cardinally equivalent, but not ordinally equivalent.

 

Thank you very very very much!  The most well-deserved  (+1) I have ever given.  You have done what I was beginning to think was impossible, you have resolved my issue completely (though I do not like the rule, I will definitely need to let it sink in more.).

 

 

Edited May 5, 2023 by Boltzmannbrain