We are forced to use forcing for almost all "hard" independence results such as: $Con(ZFC)\longrightarrow Con (ZFC+\neg CH) $. The question simply is:


**Primary Question:** Is there any "forcing free" proof for $Con(ZFC)\longrightarrow Con (ZFC+\neg CH) $ or $Con(ZF)\longrightarrow Con (ZF+\neg AC) $ or any other "hard" independence results? 


**Secondary Question:** Please list all "non simple" consistency results which have two proofs one by forcing and another without using it.



Is there any forcing free proof for hard independence results?
Krivine realizability can be used to obtain independence results over $\mathsf{ZF}$. For instance, in Krivine's paper *[Realizability algebras II : new models of ZF + DC](http://www.pps.univ-paris-diderot.fr/~krivine/articles/R_ZF.pdf)*, Logical Methods in Computer Science 8 (1:10) p. 1-28 (2012), he constructs a realizability model in which $\mathsf{AC}$ fails in a pretty strong way: there is an infinite sequence of infinite subsets of $\mathbb{R}$ strictly decreasing in cardinality.


