Borel sets with large squares

For a cardinal mu we give a sufficient condition (*)_mu (involving ranks measuring existence of independent sets) for: [(**)_mu] if a Borel set B subseteq R x R contains a mu-square (i.e. a set of the form A x A, |A|= mu) then it contains a 2^{aleph_0}-square and even a perfect square, and also for [(***)_mu] if psi in L_{omega_1, omega} has a model of cardinality mu then it has a model of cardinality continuum generated in a nice, absolute way. Assuming MA + 2^{aleph_0}> mu for transparency, those three conditions ((*)_mu, (**)_mu and (***)_mu) are equivalent, and by this we get e.g.: for all alpha= aleph_alpha => not (**)_{aleph_alpha}, and also min {mu :(*)_mu}, if <2^{aleph_0}, has cofinality aleph_1. We deal also with Borel rectangles and related model theoretic problems

The Second-order Version of Morley's Theorem on the Number of Countable Models does not Require Large Cardinals

The consistency of a second-order version of a theorem of Morley on the number of countable models was proved in arXiv:2107.07636 with the aid of large cardinals. We here dispense with them

On what I do not understand (and have something to say): Part I

This is a non-standard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdote and opinion. So the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discuss many problems, history and side references are erratic, usually kept at a minimum (``see ... '' means: see the references there and possibly the paper itself). The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The other half, concentrating on model theory, will subsequently appear

An undecidable extension of Morley's theorem on the number of countable models

We show that Morley's theorem on the number of countable models of a countable first-order theory becomes an undecidable statement when extended to second-order logic. More generally, we calculate the number of equivalence classes of $\sigma$-projective equivalence relations in several models of set theory. Our methods include random and Cohen forcing, Woodin cardinals and Inner Model Theory.Comment: 31 page