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10h

comment 
Classification of countable posets?
@Andres Caicedo Are you sure you haven't changed the meaning? I would have waited for the OP to resolve the apparent ambiguity. By the way, your version has too many betweens. 
10h

comment 
Classification of countable posets?
@AndresCaicedo How did you figure out what the OP meant? It just seemed ambiguous to me; maybe he neglected to say that the two elements were comparable, maybe he wrote poset when he meant totally ordered set. In any case, the question seems either too easy for Math Overflow, or too hard. 
13h

comment 
Classification of countable posets?
Or are you asking about totally ordered sets? In that case, your mention of "posets" is misleading, but Bjørn KjosHanssen's answer is correct. I believe this was proved by Georg Cantor in the early days of set theory. 
14h

comment 
Classification of countable posets?
I guess you mean "between each two comparable elements"? I.e., if $a\lt b$ there is an element $x$ such that $a\lt x\lt c$? (Asking for a third element between two imcomparable elements has no obvious meaning.) In other words, you want a classification of countable partial orders in which every chain is densely ordered? Those can be very complicated. 
14h

comment 
Classification of countable posets?
@AaronMeyerowitz And $[0,1]\cup(2,3]$ is isomorphic to $[0,1]$. The posted answer is correct as regards the classification of countable dense linear orders. By the way, every countable linear order is isomorphic to a subset of $\mathbb Q$. 
15h

comment 
Classification of countable posets?
I think the OP was asking about partial orders. 
Sep 25 
revised 
What is the best way to construct an Aronszajn Tree?
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Sep 25 
revised 
What is the best way to construct an Aronszajn Tree?
edited body 
Sep 25 
revised 
What is the best way to construct an Aronszajn Tree?
added 183 characters in body 
Sep 24 
answered  What is the best way to construct an Aronszajn Tree? 
Aug 28 
revised 
Coloring graph such that the coloring classes are not maximal independent sets
added 95 characters in body 
Aug 28 
answered  Coloring graph such that the coloring classes are not maximal independent sets 
Aug 23 
comment 
Who defined and who coined “module”?
That 1927 Monthly article is also the online OED's earlist citation for this sense of "module". 
Aug 22 
revised 
When does a hypergraph represent maximal independent sets?
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Aug 22 
answered  When does a hypergraph represent maximal independent sets? 
Aug 20 
comment 
Is any axiom system for sets categorical?
But a countable model of set theory can be characterized up to isomorphism by a single firstorder sentence of infinite length, right? 
Jul 31 
comment 
Proofs of the uncountability of the reals.
Wouldn't it be simpler to work with surjections instead of injections? There is a surjection from $\mathcal P(\omega)$ to $\omega_1$, there is no surjection from $\omega$ to $\omega_1$, therefore $\mathcal P(\omega)$ is uncountable. 
Jul 16 
answered  Is quasivariety generated by all perfect graphs finitely axiomatizable? 
Jul 16 
comment 
Is quasivariety generated by all perfect graphs finitely axiomatizable?
If the class is axiomatizable by universal Horn sentences, and if it's finitely axiomatizable, then it's finitely axiomatizable by Horn sentences; this follows from the compactness theorem. However, the class of perfect graphs is not closed under direct product. Let $G$ be the graph obtained by adding one more edge to the cycle $C_5$.Then $G$ is perfect, but the direct product $G\times G$ is easily seen to contain $C_5$ as an induced subgraph. so it's not perfect. Many different products of graphs are considered in graph theory; maybe the theorem you cited is about some other graph product? 
Jul 15 
revised 
Does the symmetric group on an infinite set have a minimal generating set?
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