Correlation of grammatical and semantic parameters in portuguese change-of-state verbs

Verbs denoting change of state are a semantic-syntactic universal as they share common patterns in languages of different types. One of such patterns is their ability to take part in causative-inchoative alternation when one and the same situation codified by certain language units can be seen both from the viewpoint of the actor vs recipient (transitive model) and from the viewpoint of the recipient only, provided that the recipient is the only agent in the verb’s structure (inchoative model), unlike in passive constructions. In causative-inchoative alternation situations these verbs choose one of the three alternation types: a) suppletive (matar – morrer), b) anticausative, when the inchoative meaning is codified morphologically with the help of the pronominal particle se (espantar – espantar-se) and c) labile, when the inchoative meaning is optionally marked by se without any change in grammaticality or semantics. The present paper argues that the choice of pronominal or non-pronominal form of the verb in the inchoative meaning with an inanimate subject (A janela (se) quebrou) depends on the parameter of animate / inanimate subject in the corresponding transitive construction and, to a certain extent, on the graduality inherent to the verb’s semantics (the acceptability of quantification). It is also shown that Brasilian Portuguese reveals the tendency to realize the labile type of alternation when the subject of the inchoative verb is inanimate; in Old Portuguese, on the contrary, the anticausative type (marked by se) was more frequent.

Ordinal definability and combinatorics of equivalence relations

Assume [Formula: see text]. Let [Formula: see text] be a [Formula: see text] equivalence relation coded in [Formula: see text]. [Formula: see text] has an ordinal definable equivalence class without any ordinal definable elements if and only if [Formula: see text] is unpinned. [Formula: see text] proves [Formula: see text]-class section uniformization when [Formula: see text] is a [Formula: see text] equivalence relation on [Formula: see text] which is pinned in every transitive model of [Formula: see text] containing the real which codes [Formula: see text]: Suppose [Formula: see text] is a relation on [Formula: see text] such that each section [Formula: see text] is an [Formula: see text]-class, then there is a function [Formula: see text] such that for all [Formula: see text], [Formula: see text]. [Formula: see text] proves that [Formula: see text] is Jónsson whenever [Formula: see text] is an ordinal: For every function [Formula: see text], there is an [Formula: see text] with [Formula: see text] in bijection with [Formula: see text] and [Formula: see text].

MINIMUM MODELS OF SECOND-ORDER SET THEORIES

In this article I investigate the phenomenon of minimum and minimal models of second-order set theories, focusing on Kelley–Morse set theory KM, Gödel–Bernays set theory GB, and GB augmented with the principle of Elementary Transfinite Recursion. The main results are the following. (1) A countable model of ZFC has a minimum GBC-realization if and only if it admits a parametrically definable global well order. (2) Countable models of GBC admit minimal extensions with the same sets. (3) There is no minimum transitive model of KM. (4) There is a minimum β-model of GB+ETR. The main question left unanswered by this article is whether there is a minimum transitive model of GB+ETR.

The downward directed grounds hypothesis and very large cardinals

A transitive model [Formula: see text] of ZFC is called a ground if the universe [Formula: see text] is a set forcing extension of [Formula: see text]. We show that the grounds of[Formula: see text][Formula: see text][Formula: see text] are downward set-directed. Consequently, we establish some fundamental theorems on the forcing method and the set-theoretic geology. For instance, (1) the mantle, the intersection of all grounds, must be a model of ZFC. (2) [Formula: see text] has only set many grounds if and only if the mantle is a ground. We also show that if the universe has some very large cardinal, then the mantle must be a ground.

Tame failures of the unique branch hypothesis and models of ADℝ + Θ is regular

In this paper, we show that the failure of the unique branch hypothesis ([Formula: see text]) for tame iteration trees implies that in some homogenous generic extension of [Formula: see text] there is a transitive model [Formula: see text] containing [Formula: see text] such that [Formula: see text] is regular. The results of this paper significantly extend earlier works from [Non-tame mice from tame failures of the unique branch bypothesis, Canadian J. Math. 66(4) (2014) 903–923; Core models with more Woodin cardinals, J. Symbolic Logic 67(3) (2002) 1197–1226] for tame trees.

DEFINABILITY OF SATISFACTION IN OUTER MODELS

Let M be a transitive model of ZFC. We say that a transitive model of ZFC, N, is an outer model of M if M ⊆ N and ORD ∩ M = ORD ∩ N. The outer model theory of M is the collection of all formulas with parameters from M which hold in all outer models of M (which exist in a universe in which M is countable; this is independent of the choice of such a universe). Satisfaction defined with respect to outer models can be seen as a useful strengthening of first-order logic. Starting from an inaccessible cardinal κ, we show that it is consistent to have a transitive model M of ZFC of size κ in which the outer model theory is lightface definable, and moreover M satisfies V = HOD. The proof combines the infinitary logic L∞,ω, Barwise’s results on admissible sets, and a new forcing iteration of length strictly less than κ+ which manipulates the continuum function on certain regular cardinals below κ. In the appendix, we review some unpublished results of Mack Stanley which are directly related to our topic.

Non-tame Mice from Tame Failures of the Unique Branch Hypothesis

In this paper, we show that the failure of the unique branch hypothesis (UBH) for tame trees implies that in some homogenous generic extension of V there is a transitive model M containing Ord ∪ℝ such that M ⊧ AD+ + Θ > θ0. In particular, this implies the existence (in V) of a non-tame mouse. The results of this paper significantly extend J. R. Steel's earlier results for tame trees.

Pointwise definable models of set theory

A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens V = HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are continuum many pointwise definable models of ZFC. If there is a transitive model of ZFC, then there are continuum many pointwise definable transitive models of ZFC. What is more, every countable model of ZFC has a class forcing extension that is pointwise definable. Indeed, for the main contribution of this article, every countable model of Gödel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters.

Iterates of the core model

Let N be a transitive model of ZFC such that “N ⊂ N and P(ℝ) ⊂ N. Assume that both V and N satisfy “the core model K exists.” Then KN is an iterate of K, i.e., there exists an iteration tree F on K such that F has successor length and . Moreover, if there exists an elementary embedding π: V → N then the iteration map associated to the main branch of F equals π յ K. (This answers a question of W. H. Woodin, M. Gitik, and others.) The hypothesis that P(ℝ) ⊂ N is not needed if there does not exist a transitive model of ZFC with infinitely many Woodin cardinals.

Outer models and genericity

Why is forcing the only known method for constructing outer models of set theory?If V is a standard transitive model of ZFC, then a standard transitive model W of ZFC is an outer model of V if V ⊆ W and V ∩ OR = W ∩ OR.Is every outer model of a given model a generic extension? At one point Solovay conjectured that if 0# exists, then every real that does not construct 0# lies in L[G], for some G that is generic for some forcing ℙ ∈ L. Famously, this was refuted by Jensen's coding theorem. He produced a real that is generic for an L-definable class forcing property, but does not lie in any set forcing extension of L.Beller, Jensen, and Welch in Coding the universe [BJW] revived Solovay's conjecture by asking the following question: Let a ⊆ ω be such that L[a] ⊨ “0# does not exist”. Is there ab∈ L[a] such that a ∉ L[b] and a is set generic over L[b].