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We introduce finite support iterations of symmetric systems, and use them to provide a strongly modernized proof of David Pincus’ classical result that the axiom of dependent choice is independent over $\operatorname {\mathrm {ZF}}$ with the ordering principle together with a failure of the axiom of choice.
We consider of constructing normal ultrafilters in extensions are here Easton support iterations of Prikry-type forcing notions. New ways presented. It turns out that, in contrast with other supports, seemingly unrelated measures or extenders can be involved here.
Let P be a forcing notion and $G\subseteq P$ its generic subset. Suppose that we have in $V[G]$ a $\kappa{-}$complete ultrafilter1,2W over $\kappa $. Set $U=W\cap V$.
Matrix iterative methods of solving systems of linear algebraic equations for a class of matrices which includes strictly and irreducibly diagonally dominant matrices are considered and a convergence theorem proved.
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