Annals of Pure and Applied Logic最新文献

By combining tree representation of sets with the method introduced in the previous three papers I–III [39], [35], [37] in the series, we give a new ${\Pi }_2^1$-preserving interpretation of $\mathrm{KP}{\omega }_r+\left({\Pi }_{n+2}\text{-}\mathrm{Found}\right)+\theta$ (Kripke–Platek set theory with the foundation schema restricted to ${\Pi }_{n+2}$, and augmented by θ) in ${\Sigma }_1^1\text{-}{\mathrm{AC}}_0+\left({\Pi }_{n+2}^1\text{-}\mathrm{TI}\right)+\theta$ for any ${\Pi }_2^1$ sentence θ, where the language of second order arithmetic is considered as a sublanguage of that of set theory via the standard interpretation. Thus the addition of any ${\Pi }_2^1$ theorem of $\mathrm{BI}\equiv {\Sigma }_1^1\text{-}{\mathrm{AC}}_0+\left({\Pi }_{\infty }^1\text{-}\mathrm{TI}\right)$ does not increase the consistency strength of KPω. Among such ${\Pi }_2^1$ theorems are several fixed point principles for positive arithmetical operators and ω-model reflection (the cofinal existence of coded ω-models) for theorems of BI. The reader's familiarity to the previous works I–III in the series might help, but is not necessary.

The algebraic dimension of a Polish permutation group $Q\le \mathrm{Sym}\left(N\right)$ is the size of the largest $A\subseteq N$ with the property that the orbit of every $a\in A$ under the pointwise stabilizer of $A\setminus \left\{a\right\}$ is infinite. We study the Bernoulli shift $P\curvearrowright R^N$ for various Polish permutation groups P and we provide criteria under which the P-shift is generically ergodic relative to the injective part of the Q-shift, when Q has algebraic dimension ≤n. We use this to show that the sequence of pairwise ⁎-reduction-incomparable equivalence relations defined in [18] is a strictly increasing sequence in the Borel reduction hierarchy. We also use our main theorem to exhibit an equivalence relation of pinned cardinal ${\aleph }_1^{+}$ which strongly resembles the equivalence relation of pinned cardinal ${\aleph }_1^{+}$ from [25], but which does not Borel reduce to the latter. It remains open whether they are actually incomparable under Borel reductions.

Our proofs rely on the study of symmetric models whose symmetries come from the group Q. We show that when Q is “locally finite”—e.g. when $Q=\mathrm{Aut}\left(M\right)$, where $M$ is the Fraïssé limit of a Fraïssé class satisfying the disjoint amalgamation property—the corresponding symmetric model admits a theory of supports which is analogous to that in the basic Cohen model.

We discuss some highlights of our computer-verified proof of the construction, given a countable transitive set-model M of ZFC, of generic extensions satisfying $\mathrm{ZFC}+\neg \mathrm{CH}$ and $\mathrm{ZFC}+\mathrm{CH}$. Moreover, let $R$ be the set of instances of the Axiom of Replacement. We isolated a 21-element subset $\Omega \subseteq R$ and defined $F:R\to R$ such that for every $\Phi \subseteq R$ and M-generic G, $M\models \mathrm{ZC}\cup F\text{“}\Phi \cup \Omega$ implies $M\left[G\right]\models \mathrm{ZC}\cup \Phi \cup \{\neg \mathrm{CH}\}$, where ZC is Zermelo set theory with Choice.

To achieve this, we worked in the proof assistant Isabelle, basing our development on the Isabelle/ZF library by L. Paulson and others.

Matatyahu Rubin has shown that a sharp version of Vaught's conjecture, $I(T,\omega )\in \{0,1,c\}$, holds for each complete theory of linear order $T$. We show that the same is true for each complete theory of partial order having a model in the minimal class of partial orders containing the class of linear orders and which is closed under finite products and finite disjoint unions. The same holds for the extension of the class of rooted trees admitting a finite monomorphic decomposition, obtained in the same way. The sharp version of Vaught's conjecture also holds for the theories of trees which are infinite disjoint unions of linear orders.

Given a finite lattice L that can be embedded in the recursively enumerable (r.e.) Turing degrees $〈R_T,{\le }_T〉$, it is not known how one can characterize the degrees $d\in R_T$ below which L can be embedded. Two important characterizations are of the $L_7$ and $M_3$ lattices, where the lattices are embedded below d if and only if d contains sets of “fickleness” >ω and $\ge {\omega }^{\omega }$ respectively. We work towards finding a lattice that characterizes the levels above ${\omega }^2$, the first non-trivial level after ω. We considered lattices that are as “short” in height and “narrow” in width as $L_7$ and $M_3$, but the lattices characterize also the >ω or $\ge {\omega }^{\omega }$ levels, if the lattices are not already embeddable below all non-zero r.e. degrees. We also considered upper semilattices (USLs) by removing the bottom meet(s) of some previously considered lattices, but the removals did not change the levels characterized. We discovered three lattices besides $M_3$ that also characterize the $\ge {\omega }^{\omega }$-levels. Our search for $>{\omega }^2$-candidates can therefore be reduced to the lattice-theoretic problem of finding lattices that do not contain any of the four $\ge {\omega }^{\omega }$-lattices as sublattices.

In this work we study the notions of structural and universal completeness both from the algebraic and logical point of view. In particular, we provide new algebraic characterizations of quasivarieties that are actively and passively universally complete, and passively structurally complete. We apply these general results to varieties of bounded lattices and to quasivarieties related to substructural logics. In particular we show that a substructural logic satisfying weakening is passively structurally complete if and only if every classical contradiction is explosive in it. Moreover, we fully characterize the passively structurally complete varieties of $\mathrm{MTL}$-algebras, i.e., bounded commutative integral residuated lattices generated by chains.

In this paper, we study some tree properties and their related indiscernibilities. First, we prove that SOP₂ can be witnessed by a formula with a tree of tuples holding ‘arbitrary homogeneous inconsistency’ (e.g., weak k-TP₁ conditions or other possible inconsistency configurations).

And we introduce a notion of tree-indiscernibility, which preserves witnesses of SOP₁, and by using this, we investigate the problem of (in)equality of SOP₁ and SOP₂.

Assuming the existence of a formula having SOP₁ such that no finite conjunction of it has SOP₂, we observe that the formula must witness some tree-property-like phenomenon, which we will call the antichain tree property (ATP, see Definition 4.1). We show that ATP implies SOP₁ and TP₂, but the converse of each implication does not hold. So the class of NATP theories (theories without ATP) contains the class of NSOP₁ theories and the class of NTP₂ theories.

At the end of the paper, we construct a structure whose theory has a formula having ATP, but any conjunction of the formula does not have SOP₂. So this example shows that SOP₁ and SOP₂ are not the same at the level of formulas, i.e., there is a formula having SOP₁, while any finite conjunction of it does not witness SOP₂ (but a variation of the formula still has SOP₂).

We study the properties of the constructible universe, L, over intuitionistic theories. We give an extended set of fundamental operations which is sufficient to generate the universe over Intuitionistic Kripke-Platek set theory without Infinity. Following this, we investigate when L can fail to be an inner model in the traditional sense. Namely, we show that over Constructive Zermelo-Fraenkel (even with the Power Set axiom) one cannot prove that the Axiom of Exponentiation holds in L.

Implicative algebras have been recently introduced by Miquel in order to provide a unifying notion of model, encompassing the most relevant and used ones, such as realizability (both classical and intuitionistic), and forcing. In this work, we initially approach implicative algebras as a generalization of locales, and we extend several topological-like concepts to the realm of implicative algebras, accompanied by various concrete examples. Then, we shift our focus to viewing implicative algebras as a generalization of partial combinatory algebras. We abstract the notion of a category of assemblies, partition assemblies, and modest sets to arbitrary implicative algebras, and thoroughly investigate their categorical properties and interrelationships.

This work presents a proof-theoretical and model-theoretical approach to probabilistic temporal logic. We present two novel logics; each of them extends both the language of linear time logic (LTL) and the language of probabilistic logic with polynomial weight formulas. The first logic is designed for reasoning about probabilities of temporal events, allowing statements like “the probability that A will hold in next moment is at least the probability that B will always hold” and conditional probability statements like “probability that A will always hold, given that B holds, is at least one half”, where A and B are arbitrary statements. We axiomatize this logic, provide corresponding sigma additive semantics and prove that the axiomatization is sound and strongly complete. We show that the satisfiability problem for our logic is decidable, by presenting a procedure which runs in polynomial space. We also present a logic with much richer language, in which probabilities are not attached only to temporal events, but the language allows arbitrary nesting of probability and temporal operators, allowing statements like “probability that tomorrow the chance of rain will be less than 80% is at least a half”. For this logic we prove a decidability result.