### ZF + DC + AX$_4$

by Shelah. [Sh:1005]

Archive for Math Logic, 2016

We consider mainly the following version of set theory:``ZF +
DC
and for every lambda, lambda^{aleph_0} is well ordered'', our
thesis is that this is a reasonable set theory, e.g. much can
be said. In particular, we prove that for a sequence bar delta
= <
delta_s:s in Y>, textrm {cf}(delta_s) large enough compared to
Y, we can prove the pcf theorem with minor changes (using true
cofinalities not the pseudo one). We then deduce the existence of
covering numbers and define and prove existence of truely
successor cardinals. From this we show that some
diagonalization arguments (more specifically some black boxes and
consequence) on Abelian groups. We end but showing some such
consequences hold in ZF above.

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