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This example appears in Jech's The Axiom of Choice. I am well aware of this construction, my question is whether or not this sort of behaviour (where you can replace "finite subspaces" by "subspaces of cardinality $<\kappa$" for a fixed $\kappa$) can always be found in the absence of choice.
@Andres: I know that one of the key differences between Random and Cohen reals is that Random reals are dominated by some function in the ground model, while Cohen reals are not (as you mentioned in your answer). Is this the key property for the ground model reals becoming of measure zero? (i.e. if we add one Random real, does it "nullify" (in a measure theoretic sense) the real numbers of the ground model?)
@Thierry: For the past couple of weeks I spent a lot time considering models without choice, not only I held that misconception but not once anyone corrected me about it - grad students and professors alike.
I'm not even close to know enough for a proper answer, but I know that some large cardinal axioms might not be linearly ordered. You can find a graph in Kanamori's The Higher Infinite, and see there for yourself some of the possibly-non-linear sentences.
@David: Actually it is not very hard to add one more to a countable family, the surprising part is that it carries over to the uncountable case. For example, every countable ordinal can be embedding into the rationals, but $\omega_1$ cannot be embedded into it (even without preserving the order).
In the introductory course I was TA'ing last semester we gave an exercise to define $\le$, $+$ and $\cdot$ from $a\mod b$ (when everything is zero when taken mod 0).
