Hi @LeAnhDung , it may seems weird, but are you the new user Akira?

Yes it's me @Holo

@LeAnhDung I see, can I know why?(only if you want to tell of course)

Actually, I have used up to my quota for questions. And I'm only able to ask questions in in week (you know MSE regulation rules). But I'm stuck at Linear and Complete Linear Ordering, i found it hard not to resolve my uncertainty. That's it :)

@LeAnhDung There is regulation for questions per week?? Really? I never knew

Oh ^^ Since you don't need to ask questions like me ^^

6 questions per 24h, and 50 questions per 30 days

@LeAnhDung Haha, thanks for the compliment but believe you me I do ask questions, and a lot, but I have a class to do so :)

@LeAnhDung I see

I know it's not fair to do so

It is not fair to ask in class ??

It's now a tough period of time and i hope that i will soon get over it. After that all things will become as usual.

Thank your for your sympathy :)

I'm very curious about how you know Akira is acutually me

@LeAnhDung I saw you commenting using this account to your question

You first comment this exact comment using Le Anh Dung

Good catch! You have seen that comment before I realized that mistake and deleted it

Do you know character Akira?

Akira as from the movie Akira?

Bravo Yes it is

I love that name

^^ You are in the opposite polar to me in that respect

I didn't saw the movie tho, I don't really like action shows

@LeAnhDung Akira is pretty common name in japan

But you did know that movie, that's interesting

I have a close friend who is a manga fan but he does not know that movie

This movie is too famous for me to not knowing it(I have basically 3 things I do on my free time: math, anime/manga and programming)

@LeAnhDung Btw, I hope that order theory didn't took everything from you! set theory is 10 times more interesting!

You know actually my I just study Set Theory as a stepping stone to study Analysis and Topology. I hope that I will soon finish it to get to the main courses

It seems to me that we can not understand Analysis without knowing how to build number systems

Well, let me tell you a secret, no Analysis course require from you set theory(although advance topology does)

But set theory is great by its own

For example, my favourite question ever("Is, forall $n\in\Bbb N$, there partition $P$ of $\Bbb R^+$ such that $|P|=n$ and every element in the partition is closed under addition?(Assuming AC)") is set theory one

It's hard for me to understand that. I meant how do we actually understand the theorem that every bounded sequence has a limit without knowing the completeness of $\Bbb R$

Your questions is out of my reach in the meantime :)

@LeAnhDung usually, completeness is an axiom(look here, all of those are equivalent to completeness) So no need to set theory(although it helps to intuition)

@LeAnhDung This is pretty hard, do you know what Teichmüller–Tukey lemma is?

Oh I see, we build $\Bbb R$ by axioms, not constructively

I heard that theorem some times on MSE

It seems to equivalent to AC

Yes, so I use that to show using induction

Great! I have seen a proof that requires finite character before and I stay back ^^

too much for me

It actually not that hard the moment you know what to do(well dah :)), try to use Teichmüller–Tukey lemma on the set $\mathcal{F}=\left\{ x\in\mathcal{P}(\mathbb{R}^{+})\mid\mbox{arbitrary sum of elements of $x$ does not equal an integer}\right\}$, and show that the maximal element of $(\mathcal F,\subseteq)$ and $\Bbb R^+$ minus that maximal element are both closed under addition

When I finish my textbook, i will come back to your problem and give it a try

Don't bother you any more, feel free to do your work @Holo

@LeAnhDung sure, something you can do now is "If $(X,\le)$ is well ordered, is there embedding between 2 different initial segments of $X$?", this is very important if you wish to advance more into set theory :)

@LeAnhDung Not bothering at all, my semester didn't even started so I have nothing to do :P

@LeAnhDung Well, there are 2 ways to build $\Bbb R$ and both are equal

One is from the bottom up using set theory

The other is from above down using field axioms