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New answer. For any sequence $a=\langle a_\alpha\mid\alpha<\ell(a)\rangle$, define $\Box a$ to the sequence $\langle b_\alpha\mid\alpha<\ell(a)\rangle$, where $b_\alpha=\min\{a_\beta\mid\alpha<\beta\} …

A partial order relation $\leq$ on a set $A$ is a binary relation that is reflexive, transitive, and antisymmetric. A preorder relation $\unlhd$ (also sometimes known as a quasi order or pseudo order) …

The answer is no. In the Boolean algebra $P(\omega)/\text{Fin}$ every strictly descending sequence $A_0>A_1>A_2>\cdots$ has a nonzero lower bound, by the famous construction of Hausdorff. Namely, one …

You are looking for the concept of saturated model. A model $M$ is $\kappa$-saturated if any type consisting of fewer than $\kappa$ many assertions that is consistent with the elementary diagram of $M …

The easy case occurs when $\kappa^{<\kappa}=\kappa$. Under GCH, this includes every regular cardinal. As Emil mentions in the comments, the result for this case follows from general model-theoretic co …

Yes. This is both studied by set theorists and interesting. I personally find some of the related questions below extremely interesting, connected with some very deep questions about the nature of mat …

As Sam mentioned in the comments, the answer to question 1 is yes. One can map every condition $p$ in the partial order to the lower cone, the set $S_p=\{q\in P\mid q\leq p\}$ of conditions below $p$. …

Yes, $P(\omega)/\text{fin}$ is fractal. If $A\subseteq^* B$ but not equivalent, then the interval $[A,B]$ in $P(\omega)/\text{fin}$ consists of the sets that almost contain $A$ and are almost containe …

Every such cofinal family $\mathcal{A}'$ must have size continuum. The reason is that there is an almost disjoint family $\mathcal{D}$ of size continuum, a family of infinite co-infinite sets $A\subse …

Let me mention one easy observation, which is that the order $2^{<\kappa}$ is universal for all linear orders of size $\kappa$. View the order as $\{-1,1\}^{<\kappa}$, where strings are ordered lexica …

Consider the surreal line $\langle\newcommand\No{\text{No}}\No,\leq\rangle$, in its order structure only. This is a proper class linear order, which is universal for all set-sized linear orders, as we …

The usual definition of ideal also includes the requirement that $F$ is closed under unions. But in this case, I claim that Chvatal's property holds for all infinite ideals. First, if $a\in Y\in F$, t …

Regarding the dense embedding, perhaps this is helpful. Statement 1 can be taken as a definition of density, which makes the connection with topology by means of the lower-cone topology. Theorem. Supp …

A partial order $\mathbb{B}$ is universal for a class $\cal{P}$ of partial orders if every order in $\cal{P}$ embeds order-preservingly into $\mathbb{B}$. For example, every partial order $\langle\ma …

Question. How many (isomorphism types of) finite groups of size at most n are there? What is the asymptotic growth rate? And the same question for rings, fields, graphs, partial orders, etc. Motiva …