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I'm working on the consistency of Martin's axiom and I need some help counting. Assume we are in a universe where GCH holds and $\kappa$ is a regular cardinal. How many non-isomorphic partial orders are there of size less than $\kappa$. I think the answer is $\kappa$ (indeed this is what Jech writes) and can convince myself, but it is not very clear to me how to systematically count the number of different partial orders of a certain size.

Let me know if I need to give more context, I'm asking my specific question in a general way.

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    hmm I think I figured it out. I beleive the way to do it is by counting possible chains (and thinking of them as subsets). I'd still like an expert to explain it clearly.2012-01-31
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    I'm far from an expert, but I hope my answer explains it clearly.2012-01-31

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