We give two examples. T 0 has nine countable models and a nonprincipal 1-type which contains infinitely many 2-types. T 1 has four models and an inessential extension T 2 having infinitely many models.

Two structures are said to be equimorphic if each embeds in the other. Such structures cannot be expected to be isomorphic, and in this paper we investigate the special case of linear orders, here also called chains. In particular we provide structure results for chains having less than continuum many isomorphism classes of equimorphic chains. We deduce as a corollary that any chain has either a single isomorphism class of equimorphic chains or infinitely many.

In this paper I chart the seismic shift that has occurred over the past three decades in attitudes towards the interpretation of visual images. My strategy implies the argument that the reading of visual images would appear to be an inevitability given the accelerating change of attitudes towards pictures as containers of determinate knowledge. French critical theorists (Foucault, Barthes, Derrida et. al.) dominated debate on interpretation of text and image in the 1980s, where my survey begins. Michel Foucault dismissed the (...) image (in Madness and Civilization 1959/1988) as a fascinating site for the madness of dreams but one standing outside of reasoned interpretation because of an inherent excess of meaning and deeply hidden attributes and allusions. Generally, however, when images were discussed using identifiable interpretive strategies in the 1980s the framework was a variant of semiotic analysis (Marin, Eco and Barthes – who famously, diverges from this mode in Camera Lucida). I use W. J. T. Mitchell’s Iconology: Image, Text, Ideology 1986 as a focal text from the 1980s and follow this with two of his publications on images, one from 1995 and another from 2005 to demonstrate the dramatic shift in the approach to pictures across almost three decades. Around these focal texts by Mitchell I reference other texts and trends that culminate in the more recent proliferation of texts related to visual studies and the re-emergence of aesthetics. (shrink)

With quantifier elimination and restriction of language to a binary relation symbol and constant symbols it is shown that countable complete theories having three isomorphism types of countable models are "essentially" the Ehrenfeucht example [4, $\s6$ ].

Let L be a finite relational language. The age of a structure M over L is the set of isomorphism types of finite substructures of M. We classify those ages U for which there are less than 2ω countably infinite pairwise nonisomorphic L-structures of age U.

An example is given of a complete theory with minimal models of arbitrarily large minimality rank.

It is assumed that a Kripke–Joyal semantics \({\mathcal{A} = \left\langle \mathbb{C},{\rm Cov}, {\it F},\Vdash \right\rangle}\) has been defined for a first-order language \({\mathcal{L}}\) . To transform \({\mathbb{C}}\) into a Heyting algebra \({\overline{\mathbb{C}}}\) on which the forcing relation is preserved, a standard construction is used to obtain a complete Heyting algebra made up of cribles of \({\mathbb{C}}\) . A pretopology \({\overline{{\rm Cov}}}\) is defined on \({\overline{\mathbb{C}}}\) using the pretopology on \({\mathbb{C}}\) . A sheaf \({\overline{{\it F}}}\) is made up of sections of (...) F that obey functoriality. A forcing relation \({\overline{\Vdash}}\) is defined and it is shown that \({\overline{\mathcal{A}} = \left\langle \overline{\mathbb{C}},\overline{\rm{Cov}},\overline{{\it F}}, \overline{\Vdash} \right\rangle }\) is a Kripke–Joyal semantics that faithfully preserves the notion of forcing of \({\mathcal{A}}\) . That is to say, an object a of \({\mathbb{C}Ob}\) forces a sentence with respect to \({\mathcal{A}}\) if and only if the maximal a-crible forces it with respect to \({\overline{\mathcal{A}}}\) . This reduces a Kripke–Joyal semantics defined over an arbitrary site to a Kripke–Joyal semantics defined over a site which is based on a complete Heyting algebra. (shrink)