NILPOTENCY IN UNCOUNTABLE GROUPS

2016 ◽  
Vol 103 (1) ◽  
pp. 59-69 ◽  
Author(s):  
FRANCESCO DE GIOVANNI ◽  
MARCO TROMBETTI
Keyword(s):  
Continuum Hypothesis ◽  
Large Cardinality ◽  
Generalized Continuum

The main purpose of this paper is to investigate the behaviour of uncountable groups of cardinality $\aleph$ in which all proper subgroups of cardinality $\aleph$ are nilpotent. It is proved that such a group $G$ is nilpotent, provided that $G$ has no infinite simple homomorphic images and either $\aleph$ has cofinality strictly larger than $\aleph _{0}$ or the generalized continuum hypothesis is assumed to hold. Furthermore, groups whose proper subgroups of large cardinality are soluble are studied in the last part of the paper.

The Hanf number for complete Lω1,ω-sentences (without GCH)

1974 ◽  
Vol 39 (3) ◽  
pp. 575-578 ◽  
Author(s):  
James E. Baumgartner
Keyword(s):  
Continuum Hypothesis ◽  
Order Logic ◽  
First Order Logic ◽  
Crucial Point ◽  
Basic Approach ◽  
First Order ◽  
Large Cardinality ◽  
Countable Ordinal ◽  
Generalized Continuum ◽  
Hanf Number

The Hanf number for sentences of a language L is defined to be the least cardinal κ with the property that for any sentence φ of L, if φ has a model of power ≥ κ then φ has models of arbitrarily large cardinality. We shall be interested in the language Lω1,ω (see [3]), which is obtained by adding to the formation rules for first-order logic the rule that the conjunction of countably many formulas is also a formula.Lopez-Escobar proved [4] that the Hanf number for sentences of Lω1,ω is ⊐ω1, where the cardinals ⊐α are defined recursively by ⊐0 = ℵ0 and ⊐α = Σ{2⊐β: β < α} for all cardinals α > 0. Here ω1 denotes the least uncountable ordinal.A sentence of Lω1,ω is complete if all its models satisfy the same Lω1,ω-sentences. In [5], Malitz proved that the Hanf number for complete sentences of Lω1,ω is also ⊐ω1, but his proof required the generalized continuum hypothesis (GCH). The purpose of this paper is to give a proof that does not require GCH.More precisely, we will prove the following:Theorem 1. For any countable ordinal α, there is a complete Lω1,ω-sentence σαwhich has models of power ⊐α but no models of higher cardinality.Our basic approach is identical with Malitz's. We simply use a different combinatorial fact at the crucial point.

scholarly journals Isomorphic Universality and the Number of Pairwise Nonisomorphic Models in the Class of Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Mirna Džamonja
Keyword(s):  
Banach Spaces ◽  
Continuum Hypothesis ◽  
Generalized Continuum

We develop the framework ofnatural spacesto study isomorphic embeddings of Banach spaces. We then use it to show that a sufficient failure of the generalized continuum hypothesis implies that the universality number of Banach spaces of a given density under a certain kind of positive embedding (very positive embedding) is high. An example of a very positive embedding is a positive onto embedding betweenC(K)andCLfor 0-dimensionalKandLsuch that the following requirement holds for allh≠0andf≥0inC(K): if0≤Th≤Tf, then there are constantsa≠0andbwith0≤a·h+b≤fanda·h+b≠0.

Ultraproducts which are not saturated

1967 ◽  
Vol 32 (1) ◽  
pp. 23-46 ◽  
Author(s):  
H. Jerome Keisler
Keyword(s):  
Predicate Logic ◽  
Continuum Hypothesis ◽  
First Order ◽  
Generalized Continuum ◽  
First Order Predicate Logic

In this paper we continue our study, begun in [5], of the connection between ultraproducts and saturated structures. IfDis an ultrafilter over a setI, andis a structure (i.e., a model for a first order predicate logicℒ), the ultrapower ofmoduloDis denoted byD-prod. The ultrapower is important because it is a method of constructing structures which are elementarily equivalent to a given structure(see Frayne-Morel-Scott [3]). Our ultimate aim is to find out what kinds of structure are ultrapowers of. We made a beginning in [5] by proving that, assuming the generalized continuum hypothesis (GCH), for each cardinalαthere is an ultrafilterDover a set of powerαsuch that for all structures,D-prodisα+-saturated.

On the inadequacy of inner models

1972 ◽  
Vol 37 (3) ◽  
pp. 569-571
Author(s):  
Andreas Blass
Keyword(s):  
Standard Model ◽  
Continuum Hypothesis ◽  
Axiom Of Choice ◽  
Precise Statement ◽  
Relative Consistency ◽  
Inner Models ◽  
Generalized Continuum ◽  
The Axiom Of Choice

The method of inner models, used by Gödel to prove the (relative) consistency of the axiom of choice and the generalized continuum hypothesis [2], cannot be used to prove the (relative) consistency of any statement which contradicts the axiom of constructibility (V = L). A more precise statement of this well-known fact is:(*)For any formula θ(x) of the language of ZF, there is an axiom α of the theory ZF + V ≠ L such that the relativization α(θ) is not a theorem of ZF.On p. 108 of [1], Cohen gives a proof of (*) in ZF assuming the existence of a standard model of ZF, and he indicates that this assumption can be avoided. However, (*) is not a theorem of ZF (unless ZF is inconsistent), because (*) trivially implies the consistency of ZF. What assumptions are needed to prove (*)? We know that the existence of a standard model implies (*) which, in turn, implies the consistency of ZF. Is either implication reversible?From our main result, it will follow that, if the converse of the first implication is provable in ZF, then ZF has no standard model, and if the converse of the second implication is provable in ZF, then so is the inconsistency of ZF. Thus, it is quite improbable that either converse is provable in ZF.

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