Universal models and definability

Abstract

We establish some general results on universal models in Topos Theory and show that the investigation of such models can shed light on problems of definability in Logic as well as on De Morgan's law and the law of excluded middle for Grothendieck toposes. © Copyright Cambridge Philosophical Society 2011

On the geometric theory of local MV-algebras

“…Next, we show that the radical of every model of FinRank is definable by a geometric formula {x.φ}. The fact that this is true for all the finitely presentable models of FinRank is a consequence of Corollary 3.2 [5]. To prove that it is true for general models of FinRank, we have to show that the construction of the radical A → Rad(A) commutes with filtered colimits.…”

“…Theorem 2.3 (Theorem 6.26 [5] By Theorem 2.2, the classifying topos of a quotient T ′ of a theory of presheaf type T can be represented as Sh(f.p.T-mod(Set) op , J), where J is the Gro-thendieck topology associated with T ′ . The standard method for calculating the Grothendieck topology associated with a quotient of a cartesian theory is described in D3.1.10 [17], while [2] gives a generalization which works for arbitrary theories of presheaf type.…”

“…This corresponds to condition (ii). Lastly, axiom (5) asserts that R is contained in δ(I) ∪ δ(J). Proof.…”

Topologies for intermediate logics

“…Notice that every implicationally open local operator is weakly open (indeed, the associated sheaf functor always preserves the initial object, and for any object a of , . Recall from [, Proposition 6.4] that every dense local operator is weakly open. On the other hand, the converse does not hold, since every open local operator is weakly open but not in general dense.…”

“…Recall from [, Proposition 6.2] that the Boolean or double negation topology b on is the smallest topology j on such that all the subobjects in of the form are j ‐dense, equivalently the smallest topology j on such that the equalizer of the pair of maps where f is the composite and g is the composite (where ! Ω is the unique arrow ) is j ‐dense; analogously, the De Morgan topology m on is the smallest topology j on such that all the subobjects in of the form are j ‐dense, equivalently the smallest topology j on such that the equalizer of the pair of maps is j ‐dense.…”