On a Glimm -- Effros dichotomy theorem for Souslin relations in generic universes

We prove that if every real belongs to a set generic extension of the constructible universe then every \Sigma_1^1 equivalence E on reals either admits a Delta_1^HC reduction to the equality on the set 2^{<\om_1} of all countable binary sequences, or continuously embeds E_0, the Vitali equivalence. The proofs are based on a topology generated by OD sets

Some new results on decidability for elementary algebra and geometry

We carry out a systematic study of decidability for theories of (a) real vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces, Banach spaces and metric spaces, all formalised using a 2-sorted first-order language. The theories for list (a) turn out to be decidable while the theories for list (b) are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic. We find that the purely universal and purely existential fragments of the theory of normed spaces are decidable, as is the AE fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilbert's 10th problem show that the EA fragments for metric and normed spaces and the AE fragment for normed spaces are all undecidable.Comment: 79 pages, 9 figures. v2: Numerous minor improvements; neater proofs of Theorems 8 and 29; v3: fixed subscripts in proof of Lemma 3

On a Spector ultrapower of the Solovay model

We prove that a Spector--like ultrapower extension \gN of a countable Solovay model \gM (where all sets of reals are Lebesgue measurable) is equal to the set of all sets constructible from reals in a generic extension \gM[\al] where \al is a random real over \gM. The proof involves an almost everywhere uniformization theorem in the Solovay model

On the Relative Consistency Strength of Determinacy Hypothesis

For any collection of sets of reals C, let C-DET be the statement that all sets of reals in C are determined. In this paper we study questions of the form: For given C ⊆ C', when is C'-DET equivalent, equiconsistent or strictly stronger in consistency strength than C-DET (modulo ZFC)? We focus especially on classes C contained in the projective sets

Narrow coverings of omega-product spaces

Results of Sierpinski and others have shown that certain finite-dimensional product sets can be written as unions of subsets, each of which is "narrow" in a corresponding direction; that is, each line in that direction intersects the subset in a small set. For example, if the set (omega \times omega) is partitioned into two pieces along the diagonal, then one piece meets every horizontal line in a finite set, and the other piece meets each vertical line in a finite set. Such partitions or coverings can exist only when the sets forming the product are of limited size. This paper considers such coverings for products of infinitely many sets (usually a product of omega copies of the same cardinal kappa). In this case, a covering of the product by narrow sets, one for each coordinate direction, will exist no matter how large the factor sets are. But if one restricts the sets used in the covering (for instance, requiring them to be Borel in a product topology), then the existence of narrow coverings is related to a number of large cardinal properties: partition cardinals, the free subset problem, nonregular ultrafilters, and so on. One result given here is a relative consistency proof for a hypothesis used by S. Mrowka to construct a counterexample in the dimension theory of metric spaces

Some results on embeddings of algebras, after de Bruijn and McKenzie

In 1957, N. G. de Bruijn showed that the symmetric group Sym(\Omega) on an infinite set \Omega contains a free subgroup on 2^{card(\Omega)} generators, and proved a more general statement, a sample consequence of which is that for any group A of cardinality \leq card(\Omega), Sym(\Omega) contains a coproduct of 2^{card(\Omega)} copies of A, not only in the variety of all groups, but in any variety of groups to which A belongs. His key lemma is here generalized to an arbitrary variety of algebras \bf{V}, and formulated as a statement about functors Set --> \bf{V}. From this one easily obtains analogs of the results stated above with "group" and Sym(\Omega) replaced by "monoid" and the monoid Self(\Omega) of endomaps of \Omega, by "associative K-algebra" and the K-algebra End_K(V) of endomorphisms of a K-vector-space V with basis \Omega, and by "lattice" and the lattice Equiv(\Omega) of equivalence relations on \Omega. It is also shown, extending another result from de Bruijn's 1957 paper, that each of Sym(\Omega), Self(\Omega) and End_K (V) contains a coproduct of 2^{card(\Omega)} copies of itself. That paper also gave an example of a group of cardinality 2^{card(\Omega)} that was {\em not} embeddable in Sym(\Omega), and R. McKenzie subsequently established a large class of such examples. Those results are shown to be instances of a general property of the lattice of solution sets in Sym(\Omega) of sets of equations with constants in Sym(\Omega). Again, similar results -- this time of varying strengths -- are obtained for Self(\Omega), End_K (V), and Equiv(\Omega), and also for the monoid \Rel of binary relations on \Omega. Many open questions and areas for further investigation are noted.Comment: 37 pages. Copy at http://math.berkeley.edu/~gbergman/papers is likely to be updated more often than arXiv copy Revised version includes answers to some questions left open in first version, references to results of Wehrung answering some other questions, and some additional new result

Hidden variables in quantum mechanics: Generic models, set-theoretic forcing, and the emergence of probability

The hidden-variables premise is shown to be equivalent to the existence of generic filters for algebras of commuting propositions and for certain more general propositional systems. The significance of this equivalence is interpreted in light of the theory of generic filters and boolean-valued models in set theory (the method of forcing). The apparent stochastic nature of quantum observation is derived for these hidden-variables models.Comment: 67 pages. Corrected formulas for conditional and joint probabilities per comment of J. Malle

A Generalization of Martin's Axiom

We define the $\aleph_{1.5}$ chain condition. The corresponding forcing axiom is a generalization of Martin's Axiom and implies certain uniform failures of club--guessing on $\omega_1$ that don't seem to have been considered in the literature before.Comment: 36 page