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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 Kjos-Hanssen'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?
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Sep
25
revised What is the best way to construct an Aronszajn Tree?
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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
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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 first-order 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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