ﻻ يوجد ملخص باللغة العربية
Henle, Mathias, and Woodin proved that, provided that $omegarightarrow(omega)^{omega}$ holds in a model $M$ of ZF, then forcing with $([omega]^{omega},subseteq^*)$ over $M$ adds no new sets of ordinals, thus earning the name a barren extension. Moreover, under an additional assumption, they proved that this generic extension preserves all strong partition cardinals. This forcing thus produces a model $M[mathcal{U}]$, where $mathcal{U}$ is a Ramsey ultrafilter, with many properties of the original model $M$. This begged the question of how important the Ramseyness of $mathcal{U}$ is for these results. In this paper, we show that several classes of $sigma$-closed forcings which generate non-Ramsey ultrafilters have the same properties. Such ultrafilters include Milliken-Taylor ultrafilters, a class of rapid p-points of Laflamme, $k$-arrow p-points of Baumgartner and Taylor, and extensions to a class of ultrafilters constructed by Dobrinen, Mijares and Trujillo. Furthermore, the class of Boolean algebras $mathcal{P}(omega^{alpha})/mathrm{Fin}^{otimes alpha}$, $2le alpha<omega_1$, forcing non-p-points also produce barren extensions.
We prove a strong non-structure theorem for a class of metric structures with an unstable pair of formulae. As a consequence, we show that weak categoricity (that is, categoricity up to isomorphisms and not isometries) implies severa
Fix a countable nonstandard model $mathcal M$ of Peano Arithmetic. Even with some rather severe restrictions placed on the types of minimal cofinal extensions $mathcal N succ mathcal M$ that are allowed, we still find that there are $2^{aleph_0}$ pos
In this paper, we investigate connections between structures present in every generic extension of the universe $V$ and computability theory. We introduce the notion of {em generic Muchnik reducibility} that can be used to to compare the complexity o
All known structural extensions of the substructural logic $mathsf{FL_e}$, Full Lambek calculus with exchange/commutativity, (corresponding to subvarieties of commutative residuated lattices axiomatized by ${vee, cdot, 1}$-equations) have decidable t
In cite{CGH15} we introduced TiRS graphs and TiRS frames to create a new natural setting for duals of canonical extensions of lattices. In this continuation of cite{CGH15} we answer Problem 2 from there by characterising the perfect lattices that are