Archive for Mathematical Logic最新文献

In this paper, we introduce the notion of ({textbf{K}} )-rank, where ({textbf{K}} ) is a strong amalgamation Fraïssé class. Roughly speaking, the ({textbf{K}} )-rank of a partial type is the number “copies” of ({textbf{K}} ) that can be “independently coded” inside of the type. We study ({textbf{K}} )-rank for specific examples of ({textbf{K}} ), including linear orders, equivalence relations, and graphs. We discuss the relationship of ({textbf{K}} )-rank to other ranks in model theory, including dp-rank and op-dimension (a notion coined by the first author and C. D. Hill in previous work).

Let ( {mathcal {M}}=(M, <, ldots ) ) be a weakly o-minimal structure. Assume that ( {mathcal {D}}ef({mathcal {M}})) is the collection of all definable sets of ( {mathcal {M}} ) and for any ( min {mathbb {N}} ), ( {mathcal {D}}ef_m({mathcal {M}}) ) is the collection of all definable subsets of ( M^m ) in ( {mathcal {M}} ). We show that the structure ( {mathcal {M}} ) has the strong cell decomposition property if and only if there is an o-minimal structure ( {mathcal {N}} ) such that ( {mathcal {D}}ef({mathcal {M}})={Ycap M^m: min {mathbb {N}}, Yin {mathcal {D}}ef_m({mathcal {N}})} ). Using this result, we prove that: (a) Every induced structure has the strong cell decomposition property. (b) The structure ( {mathcal {M}} ) has the strong cell decomposition property if and only if the weakly o-minimal structure ( {mathcal {M}}^*_M ) has the strong cell decomposition property. Also we examine some properties of non-valuational weakly o-minimal structures in the context of weakly o-minimal structures admitting the strong cell decomposition property.

For (nin omega ), the weak choice principle (textrm{RC}_n) is defined as follows:

For every infinite set X there is an infinite subset (Ysubseteq X) with a choice function on ([Y]^n:={zsubseteq Y:|z|=n}).

The choice principle (textrm{C}_n^-) states the following:

For every infinite family of n-element sets, there is an infinite subfamily ({mathcal {G}}subseteq {mathcal {F}}) with a choice function.

The choice principles (textrm{LOC}_n^-) and (textrm{WOC}_n^-) are the same as (textrm{C}_n^-), but we assume that the family ({mathcal {F}}) is linearly orderable (for (textrm{LOC}_n^-)) or well-orderable (for (textrm{WOC}_n^-)). In the first part of this paper, for (m,nin omega ) we will give a full characterization of when the implication (textrm{RC}_mRightarrow textrm{WOC}_n^-) holds in ({textsf {ZF}}). We will prove the independence results by using suitable Fraenkel-Mostowski permutation models. In the second part, we will show some generalizations. In particular, we will show that (textrm{RC}_5Rightarrow textrm{LOC}_5^-) and that (textrm{RC}_6Rightarrow textrm{C}_3^-), answering two open questions from Halbeisen and Tachtsis (Arch Math Logik 59(5):583–606, 2020). Furthermore, we will show that (textrm{RC}_6Rightarrow textrm{C}_9^-) and that (textrm{RC}_7Rightarrow textrm{LOC}_7^-).

In (textbf{ZF}), the well-known Fichtenholz–Kantorovich–Hausdorff theorem concerning the existence of independent families of X of size (|{mathcal {P}} (X)|) is equivalent to the following portion of the equally well-known Hewitt–Marczewski–Pondiczery theorem concerning the density of product spaces: “The product ({textbf{2}}^{{mathcal {P}}(X)}) has a dense subset of size |X|”. However, the latter statement turns out to be strictly weaker than (textbf{AC}) while the full Hewitt–Marczewski–Pondiczery theorem is equivalent to (textbf{AC}). We study the relative strengths in (textbf{ZF}) between the statement “X has no independent family of size (|{mathcal {P}}(X)|)” and some of the definitions of “X is finite” studied in Levy’s classic paper, observing that the former statement implies one such definition, is implied by another, and incomparable with some others.

A constructivisation of the cut-elimination proof for sequent calculi for classical, intuitionistic and minimal infinitary logics with geometric rules—given in earlier work by the second author—is presented. This is achieved through a procedure where the non-constructive transfinite induction on the commutative sum of ordinals is replaced by two instances of Brouwer’s Bar Induction. The proof of admissibility of the structural rules is made ordinal-free by introducing a new well-founded relation based on a notion of embeddability of derivations. Additionally, conservativity for classical over intuitionistic/minimal logic for the seven (finitary) Glivenko sequent classes is here shown to hold also for the corresponding infinitary classes.

We study (kappa )-maximal cofinitary groups for (kappa ) regular uncountable, (kappa = kappa ^{<kappa }). Revisiting earlier work of Kastermans and building upon a recently obtained higher analogue of Bell’s theorem, we show that:

1. (1)

   Any (kappa )-maximal cofinitary group has ({<}kappa ) many orbits under the natural group action of (S(kappa )) on (kappa ).

2. (2)

   If (mathfrak {p}(kappa ) = 2^kappa ) then any partition of (kappa ) into less than (kappa ) many sets can be realized as the orbits of a (kappa )-maximal cofinitary group.

3. (3)

   For any regular (lambda > kappa ) it is consistent that there is a (kappa )-maximal cofinitary group which is universal for groups of size ({<}2^kappa = lambda ). If we only require the group to be universal for groups of size (kappa ) then this follows from (mathfrak {p}(kappa ) = 2^kappa ).

A P-point ultrafilter over (omega ) is called an interval P-point if for every function from (omega ) to (omega ) there exists a set A in this ultrafilter such that the restriction of the function to A is either a constant function or an interval-to-one function. In this paper we prove the following results. (1) Interval P-points are not isomorphism invariant under (textsf{CH}) or (textsf{MA}). (2) We identify a cardinal invariant (textbf{non}^{**}({mathcal {I}}_{tiny {hbox {int}}})) such that every filter base of size less than continuum can be extended to an interval P-point if and only if (textbf{non}^{**}({mathcal {I}}_{tiny {hbox {int}}})={mathfrak {c}}). (3) We prove the generic existence of slow/rapid non-interval P-points and slow/rapid interval P-points which are neither quasi-selective nor weakly Ramsey under the assumption ({mathfrak {d}}={mathfrak {c}}) or (textbf{cov}({mathcal {B}})={mathfrak {c}}).

An appropriate framework is put forward for the construction of (lambda )-models with (infty )-groupoid structure, which we call homotopic (lambda )-models, through the use of an (infty )-category with cartesian closure and enough points. With this, we establish the start of a project of generalization of Domain Theory and (lambda )-calculus, in the sense that the concept of proof (path) of equality of (lambda )-terms is raised to higher proof (homotopy).

It is generally accepted that H. Friedman’s gap condition is closely related to iterated collapsing functions from ordinal analysis. But what precisely is the connection? We offer the following answer: In a previous paper we have shown that the gap condition arises from an iterative construction on transformations of partial orders. Here we show that the parallel construction for linear orders yields familiar collapsing functions. The iteration step in the linear case is an instance of a general construction that we call ‘Bachmann–Howard derivative’. In the present paper, we focus on the unary case, i.e., on the gap condition for sequences rather than trees and, correspondingly, on addition-free ordinal notation systems. This is partly for convenience, but it also allows us to clarify a phenomenon that is specific to the unary setting: As shown by van der Meeren, Rathjen and Weiermann, the gap condition on sequences admits two linearizations with rather different properties. We will see that these correspond to different recursive constructions of sequences.