Notes on Cardinals That Are Characterizable by a Complete (Scott) Sentence

Souldatos, Ioannis

doi:10.1215/00294527-2798727

Cited by 8 publications

(19 citation statements)

References 7 publications

(18 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…

…”

mentioning

confidence: 93%

Linear orderings and powers of characterizable cardinals

Souldatos¹

2012

Annals of Pure and Applied Logic

Self Cite

View full text Add to dashboard Cite

The current paper answers an open question of [5]. We say that a countable model M characterizes an infinite cardinal κ, if the Scott sentence of M has a model in cardinality κ, but no models in cardinality κ + . If M is linearly ordered by <, we will say that the linear ordering (M, <) characterizes κ, or that κ is characterizable by (M, <).From [2] we can deduce that if κ is characterizable, then κ + is characterizable by a linear ordering (see theorem 2.4, corollary 2.5). From [5] we know that if κ is characterizable by a dense linear ordering, then 2 κ is characterizable (see theorem 2.7).We show that if κ is homogeneously characterizable (cf. definition 2.2), then κ is characterizable by a dense linear ordering, while the converse fails (theorem 2.3).The main theorems are: 1) If κ > 2 λ is a characterizable cardinal, λ is characterizable by a dense linear ordering and λ is the least cardinal such that κ λ > κ, then κ λ is also characterizable (theorem 5.4), 2) if ℵα and κ ℵα are characterizable cardinals, then the same is true for κ ℵ α+β , for all countable β (theorem 5.5).Combining these two theorems we get that if κ > 2 ℵα is a characterizable cardinal, ℵα is characterizable by a dense linear ordering and ℵα is the least cardinal such that κ ℵα > κ, then for all β < α+ω 1 κ ℵ β is characterizable (theorem 5.7). Also if κ is a characterizable cardinal, then κ ℵα is characterizable, for all countable α (corollary 5.6). Structure of the paperThroughout the whole paper we work with countable languages L and when we refer to a dense linear ordering we mean a dense linear ordering without endpoints. The first two sections provide some background material for the characterizable cardinals and for the dense linear orderings respectively. Section 4 contains the construction that proves the following Theorem 1.1. If κ is a characterizable cardinal, then κ ℵ1 is also a characterizable cardinal.This appears as theorem 4.18 in section 4 and it will be easily generalized to λ ≥ ℵ 1 in the last section. Characterizable cardinalsThis section provides the necessary background on characterizable cardinals.

show abstract

“…

…”

mentioning

confidence: 93%

Linear orderings and powers of characterizable cardinals

Souldatos¹

2012

Annals of Pure and Applied Logic

Self Cite

View full text Add to dashboard Cite

show abstract

“…In Section 1, we explain the merger techniques for combining sentences that homogeneously characterize one cardinal (possibly in terms of another). We adapt the methods of [Sou14] to get a complete sentence with maximal models in κ and κ + .…”

Section: Structure Of the Papermentioning

confidence: 99%

“…Then we extend the trick illustrated in Theorem 1.6 to bound the number of successors of each node in the tree by κ and thus the number of paths by κ ω . The detailed axiomatization of a structure with these properties, but in a different vocabulary, by a complete sentence of L ω1,ω and the proof that it characterizes κ ω appears in [Sou14]. The extension to show κ ω is homogeneously characterized requires the further analysis of Hjorth construction in the same paper.…”

Section: Maximal Models In κ and κ ωmentioning

confidence: 99%

“…Replacing the construction that characterizes κ ω from [Sou14] with the construction that characterized κ ℵα , α < ω 1 , from [Sou12] (cf. Theorem 3) one can prove the following theorem.…”

Section: Maximal Models In κ and κ ωmentioning

confidence: 99%

“…Here the main construction is a refinement of old ideas, e.g. the characterization of κ + by a κ + -like linear order in Section 1 and the characterization of κ ω using results from [Sou14] in Section 2, combined with repeated use of sets of absolute indiscernibles. All the examples presented here are complete sentences with maximal models in more than one cardinality, which do not have arbitrarily large models.…”

Section: Open Question 42 ([Bkl14]mentioning

confidence: 99%

See 2 more Smart Citations

Complete Lω1,ω‐sentences with maximal models in multiple cardinalities

Baldwin

Souldatos

2019

Mathematical Logic Qtrly

Self Cite

View full text Add to dashboard Cite

In [BKS14] examples of incomplete sentences are given with maximal models in more than one cardinality. The question was raised whether one can find similar examples of complete sentences. In this paper we give examples of complete Lω 1 ,ωsentences with maximal models in more than one cardinality. From (homogeneous) characterizability of κ we construct sentences with maximal models in κ and in one of κ + , κ ω , 2 κ and more. Indeed, consistently we find sentences with maximal models in uncountably many distinct cardinalities.We unite ideas from [BFKL13, BKL14, Hjo02, Kni77] to find complete sentences of L ω1,ω with maximal models in multiple cardinals. There have been a number of papers finding complete sentences characterizing cardinals beginning with Baumgartner, Malitz and Knight in the 70's, refined by Laskowski and Shelah in the 90's and crowned by Hjorth's characterization of all cardinals below ℵ ω1 in the 2002. These results have been refined since. But this is the first paper finding complete sentences with maximal models in two or more cardinals. All models of these sentences have cardinality less than ω1 .

show abstract

Kurepa trees and spectra of $${\mathcal {L}}_{\omega _1,\omega }$$-sentences

Sinapova

Souldatos

2020

Arch. Math. Logic

View full text Add to dashboard Cite

Notes on Cardinals That Are Characterizable by a Complete (Scott) Sentence

Cited by 8 publications

References 7 publications

Linear orderings and powers of characterizable cardinals

Linear orderings and powers of characterizable cardinals

Complete Lω1,ω‐sentences with maximal models in multiple cardinalities

Kurepa trees and spectra of $${\mathcal {L}}_{\omega _1,\omega }$$-sentences

Contact Info

Product

Resources

About