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Some examples of Σ 1 1-universal preorders are presented, in the form of various relations of embeddability between countable coloured total orders. As an application, strengthening a theorem of , the Σ 1 1-universality of continuous embeddability for dendrites whose branch points have order 3 is obtained.
It is proved that in a suitable intuitionistic, locally classical, version of the theory ZFC deprived of the axiom of infinity, the requirement that every set be finite is equivalent to the assertion that every ordinal is a natural number. Moreover, the theory obtained with the addition of these finiteness assumptions is equivalent to a theory of… (More)
We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen's classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete… (More)
This meeting covered all important aspects of modern Set Theory , including large cardinal theory, combinatorial set theory, descriptive set theory, connections with algebra and analysis, forcing axioms and inner model theory. The presence of an unusually large number (19) of young researchers made the meeting especially dynamic.
Let R be the preorder of embeddability between countable linear orders colored with elements of Rado's partial order (a standard example of a wqo which is not a bqo). We show that R has fairly high complexity with respect to Borel reducibil-ity (e.g. if P is a Borel preorder then P ≤ B R), although its exact classification remains open.
We introduce the notion of an invariantly universal pair (S, E) where S is an analytic quasi-order and E ⊆ S ∩ S −1 is an analytic equivalence relation. This means that for any analytic quasi-order R there is a Borel set B invariant under E such that R is Borel equivalent to the restriction of S to B. We prove a general result giving a sufficient condition… (More)