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Register a new userCofinal types on $omega_2$
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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 simple cofinal types in $mathcal D_{aleph_2}$, and then proceed to show which of these types have an immediate successor in the Tukey ordering of $mathcal D_{aleph_2}$.
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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.
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 relation and $cal F$ is a family of Borel equivalence relations of non-cofinal essential complexity which together satisfy the dichotomy that for every Borel equivalence relation $E$, either $Ein {cal F}$ or $F$ is Borel reducible to $E$, then $cal F$ consists solely of smooth equivalence relations, thus the dichotomy is equivalent to a known theorem.
We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent (NIP) theories.
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 maximal independent set. 3. If in a partially ordered set all antichains are finite and all chains have size $aleph_{alpha}$, then the set has size $aleph_{alpha}$ if $aleph_{alpha}$ is regular. 4. Every partially ordered set has a cofinal well-founded subset. 5. If $G=(V_{G},E_{G})$ is a connected locally finite chordal graph, then there is an ordering $<$ of $V_{G}$ such that ${w < v : {w,v} in E_{G}}$ is a clique for each $vin V_{G}$.
We use the geometric axioms point of view to give an effective listing of the complete types of the theory $DCF_{0}$ of differentially closed fields of characteristic $0$. This gives another account of observations made in earlier papers.
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