On Projective Ordinals

We study in this paper the projective ordinals δ^1_n, where δ^1_n = sup{ξ: ξ is the length of ɑ Δ^1_n prewellordering of the continuum}. These ordinals were introduced by Moschovakis in [8] to serve as a measure of the "definable length" of the continuum. We prove first in §2 that projective determinacy implies δ^1_n 0 (the same result for odd n is due to Moschovakis). Next, in the context of full determinacy, we partly generalize (in §3) the classical fact that δ^1_1 ℵ_l and the result of Martin that δ^1_3 = ℵ_(ω + 1) by proving that δ^1_(n2+1) = λ^+_(2n+1), where λ_(2n+1) is a cardinal of cofinality ω. Finally we discuss in §4 the connection between the projective ordinals and Solovay's uniform indiscernibles. We prove among other things that ∀α (α^# exists) implies that every δ^1_n with n ≥ 3 is a fixed point of the increasing enumeration of the uniform indiscernibles

Maximal almost disjoint families, determinacy, and forcing

We study the notion of $\mathcal J$-MAD families where $\mathcal J$ is a Borel ideal on $\omega$. We show that if $\mathcal J$ is an arbitrary $F_\sigma$ ideal, or is any finite or countably iterated Fubini product of $F_\sigma$ ideals, then there are no analytic infinite $\mathcal J$-MAD families, and assuming Projective Determinacy there are no infinite projective $\mathcal J$-MAD families; and under the full Axiom of Determinacy + $V=\mathbf{L}(\mathbb{R})$ there are no infinite $\mathcal J$-mad families. These results apply in particular when $\mathcal J$ is the ideal of finite sets $\mathrm{Fin}$, which corresponds to the classical notion of MAD families. The proofs combine ideas from invariant descriptive set theory and forcing.Comment: 40 page

On a notion of smallness for subsets of the Baire space

Let us call a set A ⊆ ω^ω of functions from ω into ω σ-bounded if there is a countable sequence of functions (α_n: n Є ω)⊆ ω^ω such that every member of A is pointwise dominated by an element of that sequence. We study in this paper definability questions concerning this notion of smallness for subsets of ω^ω. We show that most of the usual definability results about the structure of countable subsets of ω^ω have corresponding versions which hold about σ-bounded subsets of ω^ω. For example, we show that every Σ_(2n+1^1 σ-bounded subset of ω^ω has a Δ_(2n+1)^1 "bound" {α_m: m Є ω} and also that for any n ≥ 0 there are largest σ-bounded Π_(2n+1)^1 and Σ_(2n+2)^1 sets. We need here the axiom of projective determinacy if n ≥ 1. In order to study the notion of σ-boundedness a simple game is devised which plays here a role similar to that of the standard ^*-games (see [My]) in the theory of countable sets. In the last part of the paper a class of games is defined which generalizes the ^*- and ^(**)-(or Banach-Mazur) games (see [My]) as well as the game mentioned above. Each of these games defines naturally a notion of smallness for subsets of ω^ω whose special cases include countability, being of the first category and σ-boundedness and for which one can generalize all the main results of the present paper

The core model induction in a choiceless context

In der Arbeit wird Woodin's Methode der Kernmodellinduktion benutzt, um die relative Konsistenz des Determiniertheitsaxiom zu zeigen. Dabei wird von einem Modell von ZF ausgegangen in dem das Auswahlaxiom nicht erfüllt ist und gezeigt, dass es ein Modell von ZF gibt in dem das Determiniertheitsaxiom gilt. Genauer werden folgende Resultate gezeigt: (1) Angenommen V ist ein Modell von "ZF + alle überabzählbaren Nachfolgerkardinalzahlen sind schwach kompakt und alle überabzählbaren Limeskardinalzahlen sind singulär". Dann gilt AD^L(R) in einer generischen Erweiterung von HOD_X. (2) Angenommen V ist ein Modell von "ZF + alle überabzählbaren Kardinalzahlen sind singulär". Dann gilt AD^L(R) in einer generischen Erweiterung von HOD_X

The Theory of Countable Analytical Sets

The purpose of this paper is the study of the structure of countable sets in the various levels of the analytical hierarchy of sets of reals. It is first shown that, assuming projective determinacy, there is for each odd n a largest countable ∏_n^1 set of reals, C_n (this is also true for n even, replacing ∏_n^1 by Σ_n^1 and has been established earlier by Solovay for n = 2 and by Moschovakis and the author for all even n > 2). The internal structure of the sets C_n is then investigated in detail, the point of departure being the fact that each C_n is a set of Δ_n^1-degrees, wellordered under their usual partial ordering. Finally, a number of applications of the preceding theory is presented, covering a variety of topics such as specification of bases, ω-models of analysis, higher-level analogs of the constructible universe, inductive definability, etc

Generalized Domination.

This thesis develops the theory of the everywhere domination relation between functions from one infinite cardinal to another. When the domain of the functions is the cardinal of the continuum and the range is the set of natural numbers, we may restrict our attention to nicely definable functions from R to N. When we consider a class of such functions which contains all Baire class one functions, it becomes possible to encode information into these functions which can be decoded from any dominator. Specifically, we show that there is a generalized Galois-Tukey connection from the appropriate domination relation to a classical ordering studied in recursion theory. The proof techniques are developed to prove new implications regarding the distributivity of complete Boolean algebras. Next, we investigate a more technical relation relevant to the study of Borel equivalence relations on R with countable equivalence classes. We show than an analogous generalized Galois-Tukey connection exists between this relation and another ordering studied in recursion theory.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113539/1/danhath_1.pd

The Set of Continuous Functions with the Everywhere Convergent Fourier Series

This paper deals with the descriptive set theoretic properties of the class EC of continuous functions with everywhere convergent Fourier series. It is shown that this set is a complete coanalytic set in C(T). A natural coanalytic rank function on EC is studied that assigns to each ƒ Є EC a countable ordinal number, which measures the "complexity" of the convergence of the Fourier series of ƒ. It is shown that there exist functions in EC (in fact even differentiable ones) which have arbitrarily large countable rank, so that this provides a proper hierarchy on EC with ω_1 distinct levels