What uncountable ordinals live in the Long Line?

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In summary, it is possible to prove that a well-ordered set is order-isomorphic to a subset of the real numbers under the usual ordering if and only if it is countable. This concept is related to whether bigger well-ordered sets can be embedded in the long line, which is constructed by taking the minimal uncountable well-ordered set and its Cartesian product with [0,1) under the dictionary order. It is possible to embed all well-ordered sets with cardinality less than or equal to aleph_1, the cardinality of the set of countable ordinals, but it becomes more complicated when AC is not allowed. The term "embed" has a technical meaning in this context, but it essentially means
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lugita15
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It is a relatively simple exercise to prove that a well-ordered set is order-isomorphic to a subset of R (under the usual ordering) if and only if it is countable. You can say that R is "too small" to contain any uncountable well-ordered sets.

So my question is, can you embed bigger well-ordered sets in the long line? For those who don't know, the long line can be constructed by taking the minimal uncountable well-ordered set (i.e. omega_1) and taking its Cartesian product with [0,1) under the dictionary order. So obviously omega_1 itself is emebeddable in the long line, just by taking the left endpoints of all the intervals [0,1). But can you embed bigger uncountable ordinals, and if so how big? I'm guessing that you may be able to embed all well-ordered sets with cardinality less than or equal to aleph_1, the cardinality of the set of countable ordinals.

Any help would be greatly appreciated.

Thank You in Advance.
 
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It seems as if you were not allowed to use AC. This is a quite unusual mathematical concepts, so some of your "easy proofs" would be interesting to see.
 
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fresh_42 said:
It seems as if you were not allowed to use AC. This is a quite unusual mathematical concepts, so some of your "easy proofs" would be interesting to see.
I think that, under reasonable interpretation, the first sentence in OP is correct (I am not good at parsing formal statements, so I am not completely sure). Consider some arbitrary set ##S \subseteq R##. Now assuming well-ordering principle (for simplicity), indeed there is always going to an ##S## that has well-orders of (only) uncountable length.

However, I think the intention might be as follows. Consider some ##S \subseteq R## and consider some specific well-order of ##S## that is of length, say ##\alpha##. I think the intention here might be that if we have any two arbitrary ordinal values ##x_1<x_2<\alpha## and we write the real numbers "occupying" the positions ##x_1,x_2## as ##r_{x_1},r_{x_2}## respectively, then the following must be true:
##r_{x_1}<r_{x_2}##
The comparison in the above line is meant to be the usual comparison relation between the real numbers.

If we assume that an arbitrary well-order of ##S## satisfies the condition in previous paragraph, then it seems to me that ##\alpha## should always be countable (unless I am making an easy mistake). I think one can just use the following two properties (from the basic properties of real numbers), with two or three steps in-between, to show that ##\alpha<\omega_1##:
----- between any two distinct real numbers there is a rational number
----- rational numbers are countableEdit:
I think there is yet another alternative way of writing the same statement. It should probably go like: Consider the linear-order formed by endowing ##R## with the usual comparison relation for real numbers. Then one can't "embed" an uncountable ordinal within this linear-order. Or perhaps(?) more casually: "one can't embed an uncountable ordinal in the real number line".

But, as far as I know, the term embed is supposed to have a technical meaning. I have the feeling that it agrees with the way I have used the term in previous paragraph, but I don't know for sure (it would actually be good to know).
 
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1. What is the Long Line?

The Long Line is a topological space that is constructed by taking an uncountable number of copies of the open unit interval and connecting them end-to-end, creating an uncountable line with a specific topology.

2. What are uncountable ordinals?

Uncountable ordinals are mathematical objects that extend the notion of counting beyond finite and countably infinite numbers. They represent the order or sequence of a set of objects that cannot be put into a one-to-one correspondence with the natural numbers.

3. How are uncountable ordinals related to the Long Line?

The Long Line is constructed using uncountable ordinals as its building blocks. Each copy of the open unit interval in the Long Line is associated with a specific uncountable ordinal, and the order of these ordinals determines the topology of the Long Line.

4. What uncountable ordinals live in the Long Line?

The Long Line contains all uncountable ordinals up to the first uncountable ordinal that is not a limit of countable ordinals, known as the first uncountable ordinal. This includes ordinals such as the aleph numbers and the beth numbers.

5. Why is it important to study the uncountable ordinals in the Long Line?

The Long Line and its associated uncountable ordinals have important applications in topology, set theory, and analysis. They provide a way to understand and study the structure of uncountable spaces, which are essential in many areas of mathematics. Additionally, the Long Line serves as a counterexample for various mathematical conjectures and can help to expand our understanding of infinite sets.

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