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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 $2^{aleph_0}$ possible theories of $(mathcal N,M)$ for such $mathcal N$s.
In this paper we start the analysis of the class $mathcal D_{aleph_2}$, the class of cofinal types of directed sets of cofinality at most $aleph_2$. We compare elements of $mathcal D_{aleph_2}$ using the notion of Tukey reducibility. We isolate some
We prove that for every Borel equivalence relation $E$, either $E$ is Borel reducible to $mathbb{E}_0$, or the family of Borel equivalence relations incompatible with $E$ has cofinal essential complexity. It follows that if $F$ is a Borel equivalence
We have observations concerning the set theoretic strength of the following combinatorial statements without the axiom of choice. 1. If in a partially ordered set, all chains are finite and all antichains are countable, then the set is countable. 2.
In set theory without the Axiom of Choice (AC), we observe new relations of the following statements with weak choice principles. 1. Every locally finite connected graph has a maximal independent set. 2. Every locally countable connected graph has a
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. Moreo