arXiv subjects

Sy-David Friedman

Publications and source records attributed to Sy-David Friedman.

22 records · Page 2Linked to original sources

arXiv2011

Projective maximal families of orthogonal measures with large continuum

We study maximal orthogonal families of Borel probability measures on $2^ω$ (abbreviated m.o. families) and show that there are generic extensions of the constructible universe $L$ in which each of the following holds: (1) There is a $Δ^1_3$-definable well order of the reals, there is a $Π^1_2$-definable m.o. family, there are no $\mathbfΣ^1_2$-definable m.o. families and $\mathfrak{b}=\mathfrak{c}=ω_3$ (in fact any reasonable value of $\mathfrak{c}$ will do). (2) There is a $Δ^1_3$-definable well order of the reals, there is a $Π^1_2$-definable m.o. family, there are no $\mathbfΣ^1_2$-definable m.o. families, $\mathfrak{b}=ω_1$ and $\mathfrak{c}=ω_2$.

math.LO↗

arXiv2011

Bounded forcing axioms and Baumgartner's conjecture

We study the spectrum of forcing notions between the iterations of $σ$-closed followed by ccc forcings and the proper forcings. This includes the hierarchy of $α$-proper forcings for indecomposable countable ordinals as well as the Axiom A forcings. We focus on the bounded forcing axioms for the hierarchy of $α$-proper forcings and connect them to a hierarchy of weak club guessing principles. We show that they are, in a sense, dual to each other. In particular, these weak club guessing principles separate the bounded forcing axioms for distinct countable indecomposable ordinals. In the study of forcings completely embeddable into an iteration of $σ$-closed followed by ccc forcing, we present an equivalent characterization of this class in terms of Baumgartner's Axiom A. This resolves a well-known conjecture of Baumgartner from the 1980's.

math.LO↗

arXiv2009

The Effective Theory of Borel Equivalence Relations

The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver and Harrington-Kechris-Louveau show that with respect to Borel reducibility, any Borel equivalence relation strictly above equality on $ω$ is above equality on ${\cal P}(ω)$, the power set of $ω$, and any Borel equivalence relation strictly above equality on the reals is above equality modulo finite on ${\cal P}(ω)$. In this article we examine the effective content of these and related results by studying effectively Borel equivalence relations under effectively Borel reducibility. The resulting structure is complex, even for equivalence relations with finitely many equivalence classes. However use of Kleene's $O$ as a parameter is sufficient to restore the picture from the noneffective setting. A key lemma is the existence of two effectively Borel sets of reals, neither of which contains the range of the other under any effectively Borel function; the proof of this result applies Barwise compactness to a deep theorem of Harrington establishing for any recursive ordinal $α$ the existence of $Π^0_1$ singletons whose $α$-jumps are Turing incomparable.

math.LO↗

arXiv2008

Large cardinals and gap-1 morasses

We present a new partial order for directly forcing morasses to exist that enjoys a significant homogeneity property. We then use this forcing in a reverse Easton iteration to obtain an extension universe with morasses at every regular uncountable cardinal, while preserving all n-superstrong (0<n<omega+1), hyperstrong and 1-extendible cardinals. In the latter case, a preliminary forcing to make the GCH hold is required. Our forcing yields morasses that satisfy an extra property related to the homogeneity of the partial order; we refer to them as mangroves and prove that their existence is equivalent to the existence of morasses. Finally, we exhibit a partial order that forces universal morasses to exist at every regular uncountable cardinal, and use this to show that universal morasses are consistent with n-superstrong, hyperstrong, and 1-extendible cardinals. This all contributes to the second author's outer model programme, the aim of which is to show that L-like principles can hold in outer models which nevertheless contain large cardinals.

math.LO↗