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4

First show that $\kappa \to (\kappa)^2_2$ holds iff $\kappa$ is inaccessible and every $\kappa$ tree has a branch - This is one of the characterizations of a weakly compact cardinal and a proof can be found in Kanamori. Now assume that every linear order on $\kappa$ has a subset of size $\kappa$ which is well ordered or reverse well ordered. Note that this ...

2

The issue is that $\sf DC$ is not the right tool for constructing measures or ultrafilters. Both, when constructed by transfinite recursion, require recursion much longer than a countable length, which really all that $\sf DC$ can give you. The tool for obtaining such objects is instead the Boolean Prime Ideal theorem, or weaker theorems like the ...

3

In many of these fake proofs, if you look closely you'll see that the infinity part plays a red herring. The same argument, supposedly, would have been that if there are two inaccessible cardinals, $\kappa<\lambda$, then $V_\lambda$ would satisfy "There exists an inaccessible cardinal", thus the theory "$\sf ZFC$+There exists an inaccessible cardinal" ...

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