We introduce a family of local ranks 
$D_Q$ depending on a finite set Q of pairs of the form 
$(varphi (x,y),q(y)),$ where 
$varphi (x,y)$ is a formula and 
$q(y)$ is a global type. We prove that in any NSOP
$_1$ theory these ranks satisfy some desirable properties; in particular, 
$D_Q(x=x)<omega $ for any finite tuple of variables x and any Q, if 
$qsupseteq p$ is a Kim-forking extension of types, then 
$D_Q(q)<D_Q(p)$ for some Q, and if 
$qsupseteq p$
In this article, we introduce a hierarchy on the class of non-archimedean Polish groups that admit a compatible complete left-invariant metric. We denote this hierarchy by 
$alpha $-CLI and L-
$alpha $-CLI where 
$alpha $ is a countable ordinal. We establish three results:
(1) G is 
$0$-CLI iff 
$G={1_G}$;
(2) G is 
$1$-CLI iff G admits a compatible complete two-sided invariant metric; and
(3) G is L-
$alpha $-CLI iff G is locally 
$alpha $-CLI, i.e., G contains an open subgroup that is 
在这篇文章中，我们介绍了一类非archimedean波兰群的层次结构，它们承认一个兼容的完全左不变度量。我们用 $alpha $-CLI 和 L-$alpha $-CLI 表示这个层次，其中 $alpha $ 是一个可数序号。我们建立了三个结果：（1）如果 $G={1_G}$ 是 $0$-CLI，则 G 是 $0$-CLI；（2）如果 G 允许一个兼容的完整双面不变度量，则 G 是 $1$-CLI；（3）如果 G 是局部 $alpha $-CLI，即 G 包含一个开放子群，而这个开放子群在 G 的局部是 $alpha $-CLI，则 G 是 L-$alpha $-CLI、随后，我们通过为$alpha <omega _1$构造非拱顶的CLI波兰群$G_alpha $和$H_alpha $来证明这个层次结构是合适的，这样的话：(1) $H_alpha $ 是 $alpha $-CLI 但不是 L-$beta $-CLI for $beta <alpha $；(2) $G_alpha $ 是 $(alpha +1)$-CLI 但不是 L-$alpha $-CLI。
Wilkie proved in 1977 that every countable model 
${mathcal M}$ of Peano Arithmetic has an elementary end extension 
${mathcal N}$ such that the interstructure lattice 
$operatorname {mathrm {Lt}}({mathcal N} / {mathcal M})$ is the pentagon lattice 
${mathbf N}_5$. This theorem implies that every countable nonstandard 
${mathcal M}$ has an elementary cofinal extension 
${mathcal N}$ such that 
$operatorname {mathrm {Lt}}({mathcal N} / {mathcal M}) cong {mathbf N}_5$. It is proved here that whenever 
${mathcal M} prec {mathcal N} models mathsf {PA}$ and 
We call an 
$alpha in mathbb {R}$ regainingly approximable if there exists a computable nondecreasing sequence 
$(a_n)_n$ of rational numbers converging to 
$alpha $ with 
$alpha - a_n < 2^{-n}$ for infinitely many 
${n in mathbb {N}}$. We also call a set 
$Asubseteq mathbb {N}$ regainingly approximable if it is c.e. and the strongly left-computable number 
$2^{-A}$ is regainingly approximable. We show that the set of regainingly approximable sets is neither closed under union nor intersection and that every c.e. Turing degree contains such a set. Furthermore, the regainingly approximable numbers lie properly between the computable and the left-computable numbers and are not closed under addition. While regainingly approximable numbers are easily seen to be i.o. K-trivial, we construct such an 
We investigate degree of satisfiability questions in the context of Heyting algebras and intuitionistic logic. We classify all equations in one free variable with respect to finite satisfiability gap, and determine which common principles of classical logic in multiple free variables have finite satisfiability gap. In particular we prove that, in a finite non-Boolean Heyting algebra, the probability that a randomly chosen element satisfies 
$x vee neg x = top $ is no larger than 
$frac {2}{3}$. Finally, we generalize our results to infinite Heyting algebras, and present their applications to point-set topology, black-box algebras, and the philosophy of logic.
The Baire algebra of a topological space X is the quotient of the algebra of all subsets of X modulo the meager sets. We show that this Boolean algebra can be endowed with a natural closure operator, resulting in a closure algebra which we denote 
$mathbf {Baire}(X)$. We identify the modal logic of such algebras to be the well-known system 
$mathsf {S5}$, and prove soundness and strong completeness for the cases where X is crowded and either completely metrizable and continuum-sized or locally compact Hausdorff. We also show that every extension of 
$mathsf {S5}$ is the modal logic of a subalgebra of 
$mathbf {Baire}(X)$, and that soundness and strong completeness also holds in the language with the universal modality.
Cummings, Foreman, and Magidor proved that Jensen’s square principle is non-compact at 
$aleph _omega $, meaning that it is consistent that 
$square _{aleph _n}$ holds for all 
$n<omega $ while 
$square _{aleph _omega }$ fails. We investigate the natural question of whether this phenomenon generalizes to singulars of uncountable cofinality. Surprisingly, we show that under some mild 
${{mathsf {PCF}}}$-theoretic hypotheses, the weak square principle 
$square _kappa ^*$ is in fact compact at singulars of uncountable cofinality.
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