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3 Reports on Mathematical Logic

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Invariant universality for projective planes

Paolini Gianluca

Reports on Mathematical Logic

2023

Vol. 58

15--27

We continue the work of [1, 2, 3] by analyzing the equivalence relation of bi-embeddability on various classes of countable planes, most notably the class of countable non-Desarguesian projective planes. We use constructions of the author from [13] to show that these equivalence relations are invariantly universal, in the sense of [3], and thus in particular complete analytic. We also introduce a new kind of Borel reducibility relation for standard Borel G-spaces, which requires the preservation of stabilizers, and explain its connection with the notion of full embeddings commonly considered in category theory.

Some results on polish groups

Paolini Gianluca, Shelah Saharon

Reports on Mathematical Logic

2020

Vol. 55

61--71

We prove that no quantifier-free formula in the language of group theory can define the ℵ1-half graph in a Polish group, thus generalising some results from [6]. We then pose some questions on the space of groups of automorphisms of a given Borel complete class, and observe that this space must contain at least one uncountable group. Finally, we prove some results on the structure of the group of automorphisms of a locally finite group: firstly, we prove that it is not the case that every group of automorphisms of a graph of power λ is the group of automorphism of a locally finite group of power λ; secondly, we conjecture that the group of automorphisms of a locally finite group of power λ has a locally finite subgroup of power λ, and reduce the problem to a problem on p-groups, thus settling the conjecture in the case λ = ℵ0.

A new ω-stable plane

Paolini Gianluca

Reports on Mathematical Logic

2020

Vol. 55

87--111

We use a variation on Mason’s α-function as a pre-dimension function to construct a not one-based ω-stable plane P (i.e. a simple rank 3 matroid) which does not admit an algebraic representation (in the sense of matroid theory) over any field. Furthermore, we characterize forking in T h(P), we prove that algebraic closure and intrinsic closure coincide in T h(P), and we show that T h(P) fails weak elimination of imaginaries, and has Morley rank ω.