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.

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...

As shown by Telgársky and Scheepers, winning strategies in the Menger game characterize $\sigma$-compactness amongst metrizable spaces. This is improved by showing that winning Markov strategies in the Menger game characterize $\sigma$-compactness amongst regular spaces, and that winning strategies may be improved to winning Markov strategies in second-countable spaces. An investigation of 2-Markov strategies introduces a new topological property between $\sigma$-compact and Menger spaces.

Under some very strong set-theoretic hypotheses, hereditarily normal spaces (also referred to as T₅ spaces) that are locally compact and hereditarily collectionwise Hausdorff can have a highly simplified structure. This paper gives a structure theorem (Theorem 1) that applies to all such ω₁-compact spaces and another (Theorem 4) to all such spaces of Lindelöf number ≤ ℵ₁. It also introduces an axiom (Axiom F) on crowding of functions, with consequences (Theorem 3) for the crowding of countably compact...

Lattice-valued possibilistic measures, conceived and developed in more detail by G. De Cooman in 1997 [2], enabled to apply the main ideas on which the real-valued possibilistic measures are founded also to the situations often occurring in the real world around, when the degrees of possibility, ascribed to various events charged by uncertainty, are comparable only quantitatively by the relations like “greater than” or “not smaller than”, including the particular cases when such degrees are not...