Annals of Pure and Applied Logic最新文献

We show that $\mathrm{PFA}$ (Proper Forcing Axiom) implies that adding any number of Cohen subsets of ω will not add an ${\omega }_2$-Aronszajn tree or a weak ${\omega }_1$-Kurepa tree, and moreover no σ-centered forcing can add a weak ${\omega }_1$-Kurepa tree (a tree of height and size ${\omega }_1$ with at least ${\omega }_2$ cofinal branches). This partially answers an open problem whether ccc forcings can add ${\omega }_2$-Aronszajn trees or ${\omega }_1$-Kurepa trees (with $\neg {\square }_{{\omega }_1}$ in the latter case).

We actually prove more: We show that a consequence of $\mathrm{PFA}$, namely the guessing model principle, $\mathrm{GMP}$, which is equivalent to the ineffable slender tree property, $\mathrm{ISP}$, is preserved by adding any number of Cohen subsets of ω. And moreover, $\mathrm{GMP}$ implies that no σ-centered forcing can add a weak ${\omega }_1$-Kurepa tree (see Section 2.1 for definitions).

For more generality, we study variations of the principle $\mathrm{GMP}$ at higher cardinals and the indestructibility consequences they entail, and as applications we answer a question of Mohammadpour about guessing models at weakly but not strongly inaccessible cardinals and show that there is a model in which there are no weak ${\aleph }_{\omega +1}$-Kurepa trees and no ${\aleph }_{\omega +2}$-Aronszajn trees.

We study a strengthening of the notion of a universally meager set and its dual counterpart that strengthens the notion of a universally null set.

We say that a subset A of a perfect Polish space X is countably perfectly meager (respectively, countably perfectly null) in X, if for every perfect Polish topology τ on X, giving the original Borel structure of X, A is covered by an $F_{\sigma }$-set F in X with the original Polish topology such that F is meager with respect to τ (respectively, for every finite, non-atomic, Borel measure μ on X, A is covered by an $F_{\sigma }$-set F in X with $\mu \left(F\right)=0$).

We prove that if $2^{{\aleph }_0}\le {\aleph }_2$, then there exists a universally meager set in $2^N$ which is not countably perfectly meager in $2^N$ (respectively, a universally null set in $2^N$ which is not countably perfectly null in $2^N$).

We use a second-order analogy ${\mathrm{PRA}}^2$ of $\mathrm{PRA}$ to investigate the proof-theoretic strength of theorems in countable algebra, analysis, and infinite combinatorics. We compare our results with similar results in the fast-developing field of primitive recursive (‘punctual’) algebra and analysis, and with results from ‘online’ combinatorics. We argue that ${\mathrm{PRA}}^2$ is sufficiently robust to serve as an alternative base system below ${\mathrm{RCA}}_0$ to study the proof-theoretic content of theorems in ordinary mathematics. (The most popular alternative is perhaps ${\mathrm{RCA}}_0^{⁎}$.) We discover that many theorems that are known to be true in ${\mathrm{RCA}}_0$ either hold in ${\mathrm{PRA}}^2$ or are equivalent to ${\mathrm{RCA}}_0$ or its weaker (but natural) analogy $2^N$-${\mathrm{RCA}}_0$ over ${\mathrm{PRA}}^2$. However, we also discover that some standard mathematical and combinatorial facts are incomparable with these natural subsystems.

Kunen's theorem that assuming the Axiom of Choice there are no Reinhardt cardinals is a key milestone in the program to understand large cardinal axioms. But this theorem is not the end of a story, rather it is the beginning.

The article studies two forms of responsibility in the setting of strategic games with imperfect information. They are referred to as seeing-to-it responsibility and counterfactual responsibility. It shows that counterfactual responsibility is definable through seeing-to-it, but not the other way around. The article also proposes a sound and complete bimodal logical system that describes the interplay between the seeing-to-it modality and the individual ex ante knowledge modality.

The standard comparison lemma of inner model theory is deficient, in that it does not in general produce a comparison of all the relevant inputs. How two mice compare can depend upon which iteration strategies are used to compare them. We shall outline here a method for comparing iteration strategies that removes this defect.

We introduce a general framework of generalized tree forcings, GTF for short, that includes the classical tree forcings like Sacks, Silver, Laver or Miller forcing. Using this concept we study the cofinality of the ideal $I\left(Q\right)$ associated with a GTF Q. We show that if for two GTF's $Q_0$ and $Q_1$ the consistency of $\mathrm{add}\left(I\right(Q_0\left)\right)<\mathrm{add}\left(I\right(Q_1\left)\right)$ holds, then we can obtain the consistency of $\mathrm{cof}\left(I\right(Q_1\left)\right)<\mathrm{cof}\left(I\right(Q_0\left)\right)$. We also show that $\mathrm{cof}\left(I\right(Q\left)\right)$ can consistently be any cardinal of cofinality larger than the continuum.

“Eggleston's dichotomy” is a “one of a kind” unique observation which broadly tells us that the characterized subgroups of the circle group (characterized by a sequence of positive integers $\left(a_n\right)$) are either countable or of cardinality $c$ depending on the asymptotic behavior of the sequence of the ratios $\frac{a_n}{a_{n-1}}$. One should note that these subgroups are generated by using the notion of usual convergence which is nothing but a special case of the more general notion of ideal convergence for the ideal Fin. It has been recently established that “Eggleston's dichotomy” fails in the case of modified versions of characterized subgroups when the ideal Fin is replaced by the natural density ideal $I_d$, or more generally, by ideals which are now known as simple density and modular simple density ideals. As all the ideals mentioned above are analytic P-ideals, a natural question arises as to whether one can isolate some appropriate property of ideals which enforces the dichotomy or the failure of it. In this article we are able to isolate that particular feature of an ideal and come out with a new class of ideals which we call, “strongly non-translation invariant ideals” (in short snt-ideals). In particular, we are able to establish that for a sequence of positive integers $\left(a_n\right)$, be it arithmetic or arising from the continued fraction expansion of an irrational number:

• (i)

  For non-snt analytic P ideals, the size of the corresponding characterized subgroups is always $c$ even if the sequence $\left(a_n\right)$ is b-bounded (i.e. the sequence of the ratios $\frac{a_n}{a_{n-1}}$ is bounded) which implies the breaking down of “Eggleston's dichotomy”.

• (ii)

  For snt analytic P ideals, the corresponding characterized subgroups are always countable if the sequence $\left(a_n\right)$ is b-bounded which means “Eggleston's dichotomy” holds.

We give four different independence relations on any exponential field. Each is a canonical independence relation on a suitable Abstract Elementary Class of exponential fields, showing that two of these are NSOP₁-like and non-simple, a third is stable, and the fourth is the quasiminimal pregeometry of Zilber's exponential fields, previously known to be stable (and uncountably categorical). We also characterise the fourth independence relation in terms of the third, strong independence.

If $B$ is a relational structure, define $P\left(B\right)$ the partial order of all substructures of $B$ that are isomorphic to it. Improving a result of Kurilić and the second author, we prove that if $R$ is the random graph, then $P\left(R\right)$ is forcing equivalent to $S⁎\dot{R}$, where $S$ is Sacks forcing and $\dot{R}$ is an ω-distributive forcing that is not forcing equivalent to a σ-closed one. We also prove that ${P(H}_3)$ is forcing equivalent to a σ-closed forcing, where $H_3$ is the generic triangle-free graph.