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Using a modification of the invariant Jensen forcing of [11] , we define a model of ZFC, in which, for a given n ≥ 3 , there exists a lightface Π n 1 -set of reals, which is a E 0 -equivalence class, hence a countable set, and which does not contain any OD element, while every non-empty countable Σ n 1 -set of reals is constructible, hence contains only OD reals.

We make use of a finite support product of $\omega_1$ clones of the Jensen minimal $\varPi^1_2$ singleton forcing to obtain a model of ZFC in which every non-empty lightface analytically definable set of reals contains a lightface analytically definable real (the full basis theorem), but there is no analytically definable wellordering of the continuum.

We make use of a finite support product of the Jensen minimal Π 2 1 singleton forcing to define a model in which Π 2 1 uniformization fails for a set with countable cross-sections. We also define appropriate submodels of the same model in which separation fails for Π 3 1 .

The aim of this paper is to demonstrate that several non-rigorous methods of mathematical reasoning in the field of divergent series, mostly related to the Euler and Hutton transforms, may be developed in a correct and consistent way by methods of the grossone analysis.

In the early years of set theory, Du Bois Reymond introduced a vague notion of infinitary pantachie meant to symbolize an infinity bigger than the infinity of real numbers. Hausdorff reformulated this concept rigorously as a maximal chain (a linearly ordered subset) in a partially ordered set of certain type, for instance, the set N N under eventual domination. Hausdorff proved the existence of a pantachy in any partially ordered set, using the axiom of choice AC. We show in this note that the pantachy existence theorem fails in the absense of AC, and moreover, even if AC is assumed, hence pantachies do exist, one may not be able to come up with an individual, effectively defined example of a pantachy.

We prove that if E is an equivalence relation Borel reducible to E 1 × E 3 then either E is Borel reducible to the equality of countable sets of reals or E 1 is Borel reducible to E . The “either” case admits further strengthening.

Following a research line suggested by Ilijas Farah, we prove that for any abelian Polish σ -compact group H there exists an F σ Radon–Nikodym ideal, that is, an ideal Z ⊆ P ( N ) together with a Borel Z -approximate homomorphism f : H → H N which is not Z -approximable by a continuous true homomorphism g : H → H N .

In this paper, we investigate refined definition of random sequences. Classical definitions (Martin-Löf tests of randomness, uncompressibility, unpredictability, or stochasticness) make use of the notion of algorithm. We present alternative definitions based on set theory and explain why they depend on the model of ZFC that is considered. We also present a possible generalization of the definition when small infinite regularities are allowed.

We prove that any Borel Abelian ordered group B , having a countable subgroup G as the largest convex subgroup, and such that the quotient B/G is order isomorphic to R , the reals, is Borel group-order isomorphic to the product R ×G , ordered lexicographically. As a main ingredient of this proof, we show, answering a question of D. Marker, that all Borel cocycles R 2 → Z are Borel coboundaries. A Borel classification theorem for Borel ordered ccc groups is proved.

We prove that if $\leq$ is an analytic partial order then either $\leq$ can be extended to a (boldface) $\Delta^1_2$ linear order similar to an antichain in $2^{