We prove the following theorem: Given a⊆ω and $1\le \alpha <{\omega }_1^{CK}$, if for some $\eta <{\aleph }_1$ and all u ∈ WO of length η, a is ${\Sigma }_{\alpha }^0\left(u\right)$, then a is ${\Sigma }_{\alpha }^0$. We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: ${\Sigma }_1^1$-Turing-determinacy implies the existence of $0$.

We investigate when two orthogonal families of sets of integers can be separated if one of them is analytic.

We prove that there is no maximum element, under Borel reducibility, in the class of analytic partial orders and in the class of analytic oriented graphs. We also provide a natural jump operator for these two classes.

In this note we give some new characterizations of distributivity of a nearlattice and we study annihilator-preserving congruence relations.

The properties of deductive systems in Hilbert algebras are treated. If a Hilbert algebra $H$ considered as an ordered set is an upper semilattice then prime deductive systems coincide with meet-irreducible elements of the lattice $DedH$ of all deductive systems on $H$ and every maximal deductive system is prime. Complements and relative complements of $DedH$ are characterized as the so called annihilators in $H$.

We introduce the concepts of an annihilator and a relative annihilator of a given subset of a BCK-algebra $𝒜$. We prove that annihilators of deductive systems of BCK-algebras are again deductive systems and moreover pseudocomplements in the lattice $𝒟\left(A\right)$ of all deductive systems on $𝒜$. Moreover, relative annihilators of $C\in 𝒟\left(A\right)$ with respect to $B\in 𝒟\left(A\right)$ are introduced and serve as relative pseudocomplements of $C$ w.r.t. $B$ in $𝒟\left(A\right)$.

A new ⋄-like principle ${\diamond }_{}$ consistent with the negation of the Continuum Hypothesis is introduced and studied. It is shown that $\neg {\diamond }_{}$ is consistent with CH and that in many models of = ω₁ the principle ${\diamond }_{}$ holds. As ${\diamond }_{}$ implies that there is a MAD family of size ℵ₁ this provides a partial answer to a question of J. Roitman who asked whether = ω₁ implies = ω₁. It is proved that ${\diamond }_{}$ holds in any model obtained by adding a single Laver real, answering a question of J. Brendle who asked whether = ω₁ in such models....

It is consistent that $${\aleph }_1<\mathrm{add}\left(𝒩\right)<\mathrm{add}\left(ℳ\right)=𝔟<\mathrm{cov}\left(𝒩\right)<\mathrm{non}\left(ℳ\right)<\mathrm{cov}\left(ℳ\right)=2^{{\aleph }_0}.$$ Assuming four strongly compact cardinals, it is consistent that $$\begin{array}{ccc}\hfill {\aleph }_1& <\mathrm{add}\left(𝒩\right)<\mathrm{add}\left(ℳ\right)=𝔟<\mathrm{cov}\left(𝒩\right)<\mathrm{non}\left(ℳ\right)\hfill & \hfill <\mathrm{cov}\left(ℳ\right)<\mathrm{non}\left(𝒩\right)<\mathrm{cof}\left(ℳ\right)=𝔡<\mathrm{cof}\left(𝒩\right)<2^{{\aleph }_0}.\end{array}$$

Shelah’s pcf theory describes a certain structure which must exist if ${\aleph }_{\omega }$ is strong limit and $2^{{\aleph }_{\omega }}>{\aleph }_{{\omega }_1}$ holds. Jech and Shelah proved the surprising result that this structure exists in ZFC. They first give a forcing extension in which the structure exists then argue that by some absoluteness results it must exist anyway. We reformulate the statement to the existence of a certain partially ordered set, and then we show by a straightforward, elementary (i.e., non-metamathematical) argument that such partially...