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Consider the set of functions from $\mathbb{N}\to\mathbb{N}$. We can impose a partial order on this set by saying that $f>g$ if $f(n)>g(n)$ for all sufficiently large $n$. By a diagonalization argument, it can be shown that one can embed $\omega_1$, the first uncountable ordinal in this Poset. Since the set of functions from $\mathbb{N}\to \mathbb{N}$ has the cardinality of $\mathbb{R}$ it is clear that one can not embed $\omega_2$ if the continuum hypothesis is true. In general what is the smallest ordinal one can not embed. In particular, is there an embedding of $\omega_1+1$ in general?

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The smallest such ordinal is $\omega_2$. I do not want to type much so I'll give you an excellent reference: Problem 24, Chapter 11, P. Komjath and V. Totik - Problems and theorems in classical set theory.

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    $\begingroup$ Maybe add some details, references, some justification except taking your word for it? :-) $\endgroup$
    – Asaf Karagila
    Apr 24, 2014 at 4:09
  • $\begingroup$ Sorry, you are right. I was being lazy to type the answer but I just found a reference. Although I think this should still count as a good exercise. $\endgroup$
    – hot_queen
    Apr 24, 2014 at 4:13
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    $\begingroup$ I made them up, of course. But it's fine, you can just take my word for it! :-) $\endgroup$
    – Asaf Karagila
    Apr 24, 2014 at 4:33
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    $\begingroup$ You are off by .11%. Take my word for it. $\endgroup$
    – hot_queen
    Apr 24, 2014 at 4:46
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    $\begingroup$ The following paper of Farah has many interesting results: matwbn.icm.edu.pl/ksiazki/fm/fm151/fm15115.pdf $\endgroup$
    – hot_queen
    Apr 26, 2014 at 1:16

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