Annals of Pure and Applied Logic最新文献

Shelah showed that the existence of free subsets over internally approachable subalgebras follows from the failure of the PCF conjecture on intervals of regular cardinals. We show that a stronger property called the Approachable Bounded Subset Property can be forced from the assumption of a cardinal λ for which the set of Mitchell orders $\left\{o\right(\mu \left)\right|\mu <\lambda \}$ is unbounded in λ. Furthermore, we study the related notion of continuous tree-like scales, and show that such scales must exist on all products in canonical inner models. We use this result, together with a covering-type argument, to show that the large cardinal hypothesis from the forcing part is optimal.

In this paper, we unify the study of classical and non-classical algebra-valued models of set theory, by studying variations of the interpretation functions for = and ∈. Although, these variations coincide with the standard interpretation in Boolean-valued constructions, nonetheless they extend the scope of validity of $\mathrm{ZF}$ to new algebra-valued models. This paper presents, for the first time, non-trivial paraconsistent models of full $\mathrm{ZF}$. Moreover, due to the validity of Leibniz's law in these structures, we will show how to construct proper models of set theory by quotienting these algebra-valued models with respect to equality, modulo the filter of the designated truth-values.

We make use of a finite support product of the Jensen-type forcing notions to define a model of the set theory $\text{ZFC}$ in which, for a given $n\ge 3$, there exists a good lightface ${\Delta }_n^1$ well-ordering of the reals but there are no any (not necessarily good) well-orderings in the boldface class ${\Delta }_{n-1}^1$.

We determine sufficient structure for an elementary topos to emulate Nelson's Internal Set Theory in its internal language, and show that any topos satisfying the internal axiom of choice occurs as a universe of standard objects and maps. This development allows one to employ the proof methods of nonstandard analysis (transfer, standardisation, and idealisation) in new environments such as toposes of G-sets and Boolean étendues.

The classical homomorphism preservation theorem, due to Łoś, Lyndon and Tarski, states that a first-order sentence φ is preserved under homomorphisms between structures if, and only if, it is equivalent to an existential positive sentence ψ. Given a notion of (syntactic) complexity of sentences, an “equi-resource” homomorphism preservation theorem improves on the classical result by ensuring that ψ can be chosen so that its complexity does not exceed that of φ.

We describe an axiomatic approach to equi-resource homomorphism preservation theorems based on the notion of arboreal category. This framework is then employed to establish novel homomorphism preservation results, and improve on known ones, for various logic fragments, including first-order, guarded and modal logics.

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.