Let *V* be a finite relational vocabulary in which no symbol has arity greater than 2. Let be countable *V*-structure which is homogeneous, simple and 1-based. The first main result says that if is, in addition, primitive, then it is strongly interpretable in a random structure. The second main result, which generalizes the first, implies (without the assumption on primitivity) that if is “coordinatized” by a set with SU-rank 1 and there is no definable (without parameters) nontrivial equivalence relation on *M* with only finite classes, then is strongly interpretable in a random structure.

A generic extension of the constructible universe by reals is defined, in which the union of -classes of *x* and *y* is a lightface set, but neither of these two -classes is separately ordinal-definable.

We study finite ℓ-colourable structures with an underlying pregeometry. The probability measure that is used corresponds to a process of generating such structures by which colours are first randomly assigned to all 1-dimensional subspaces and then relationships are assigned in such a way that the colouring conditions are satisfied but apart from this in a random way. We can then ask what the probability is that the resulting structure, where we now forget the specific colouring of the generating process, has a given property. With this measure we get the following results: (1) A zero-one law. (2) The set of sentences with asymptotic probability 1 has an explicit axiomatisation which is presented. (3) There is a formula (not directly speaking about colours) such that, with asymptotic probability 1, the relation “there is an ℓ-colouring which assigns the same colour to *x* and *y*” is defined by . (4) With asymptotic probability 1, an ℓ-colourable structure has a unique ℓ-colouring (up to permutation of the colours).

We show that a miniaturised version of Maclagan's theorem on monomial ideals is equivalent to and classify a phase transition threshold for this theorem. This work highlights the combinatorial nature of Maclagan's theorem.

]]>We show that the analogues of the embedding theorems of [3], proved for the countable models of set theory, do not hold when extended to the uncountable realm of ω_{1}-like models of set theory. Specifically, under the ⋄ hypothesis and suitable consistency assumptions, we show that there is a family of many ω_{1}-like models of , all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive ω_{1}-like model of that does not embed into its own constructible universe; and there can be an ω_{1}-like model of whose structure of hereditarily finite sets is not universal for the ω_{1}-like models of set theory.

We construct four models containing one supercompact cardinal in which level by level equivalence between strong compactness and supercompactness and level by level inequivalence between strong compactness and supercompactness are precisely controlled at each non-supercompact measurable cardinal. In these models, no cardinal κ is -supercompact, where is the least inaccessible cardinal greater than κ.

]]>We characterise the collections of infinite binary sequences that, when *barred* by a set of finite binary sequences, are also *barred* by a finite subset of such a set.

In this paper is used to denote Jensen's modification of Quine's ‘new foundations’ set theory () fortified with a type-level pairing function but without the axiom of choice. The axiom is the variant of the axiom of counting which asserts that no finite set is smaller than its own set of singletons. This paper shows that proves the consistency of the simple theory of types with infinity (). This result implies that proves that consistency of , and that proves the consistency of .

]]>Let *E* be a subset of . A linear extension operator is a linear map that sends a function on *E* to its extension on some superset of *E*. In this paper, we show that if *E* is a semi-algebraic or subanalytic subset of , then there is a linear extension operator such that is semi-algebraic (subanalytic) whenever *f* is semi-algebraic (subanalytic).

Let be a weakly o-minimal structure with the strong cell decomposition property. In this note, we show that the canonical o-minimal extension of is the unique prime model of the full first order theory of over any set . We also show that if two weakly o-minimal structures with the strong cell decomposition property are isomorphic then, their canonical o-minimal extensions are isomorphic too. Finally, we show the uniqueness of the prime models in a complete weakly o-minimal theory with prime models.

]]>We show how to interpret weak Kőnig's lemma in some recently defined theories of nonstandard arithmetic in all finite types. Two types of interpretations are described, with very different verifications. The celebrated conservation result of Friedman's about weak Kőnig's lemma can be proved using these interpretations. We also address some issues concerning the collecting of witnesses in herbrandized functional interpretations.

]]>We present a general framework for forcing on ω_{2} with finite conditions using countable models as side conditions. This framework is based on a method of comparing countable models as being membership related up to a large initial segment. We give several examples of this type of forcing, including adding a function on ω_{2}, adding a nonreflecting stationary subset of , and adding an ω_{1}-Kurepa tree.

We show that it is consistent, relative to ω many supercompact cardinals, that the super tree property holds at for all but there are weak square and a very good scale at .

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