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Define the partial ordering ≤ on the Cantor space ω 2 by x≤y iff ∀nx(n)≤y(n) (this corresponds to the subset relation on the power set of ω). A set A⊆ ω 2 is monotone reducible to a set B⊆ ω 2 iff there is a monotone (i.e., x≤y⇒ f(x)≤f(y)) continuous function f: ω 2→ ω 2 such that x∈A iff f(x)∈B. In this paper, we study the relation of monotone reducibility, with emphasis on two topics: (1) the similarities and differences between monotone reducibility on monotone sets (i.e., sets closed upward under ≤) and Wadge reducibility on arbitrary sets; and (2) the distinction (or lack thereof) between ‘monotone’ and ‘positive’, where ‘positive’ means roughly ‘a priori monotone’ but is only defined in certain specific cases. (For example, a Σ 2 0 -positive set is a countable union of countable intersections of monotone clopen sets.) Among the main results are the following: Each of the six lowest Wadge degrees contains one or two monotone degrees (of monotone sets), while each of the remaining Wadge degrees contains uncountably many monotone degrees (including uncountable antichains and descending chains); and, although ‘monotone’ and ‘positive’ coincide in a number of cases, there are classes of monotone sets which do not match any notion of ‘positive’.

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We show that a first category homogeneous zero-dimensional Borel set X can be embedded in (ω) as an ideal on ω if and only if X is homeomorphic to X × X if and only if X is Wadge-equivalent to X × X. Furthermore, we determine the Wadge classes of such X, thus giving a complete picture of the possible descriptive complexity of Borel ideals on ω. We also discuss the connection with ideals of compact sets.

For every μ < ω1, let Iμ be the ideal of all sets S⊆ ωμ whose order type is <ωμ. Ifμ = 1, then I1 is simply the ideal of all finite subsets of ω, which is known to be Σ02-complete. We show that for every μ < ω1, Iμ is Σ02μ-complete. As corollaries to this theorem, we prove that the set WOωμ of well orderings R⊆ω × ω of order type <ωμ is Σ02μ-complete, the set LPμ of linear orderings R⊆ ω × ωthat have a μ-limit point is Σ02μ+1-complete. Similarly, we determine the exact complexity of the set LTμ of trees T⊆ <ωω of Luzin height <μ, the set WRμ of well-founded partial orderings of height <μ, the set LRμ of partial orderings of Luzin height <μ, the set WFμ of well-founded trees T⊆ <ωω of height <μ(the latter is an old theorem of Luzin). The proofs use the notions of Wadge reducibility and Wadge games. We also present a short proof to a theorem of Luzin and Garland about the relation between the height of ‘the shortest tree’ representing a Borel set and the complexity of the set.

For each countable ordinal £ and pair (A0, AX) of disjoint analytic subsets of 2W, we define a closed game Jg(A0, Ax) and a complete (formula presented) subset Hgof 2W such that (i) a winning strategy for player I constructs a 2° set separating A0 from Ax\ and (ii) a winning strategy for player II constructs a continuous map (formula presented) with (formula presented). Applications of this construction include: A proof in second order arithmetics of the statement “every IIɸnon (formula presented) set is ng-complete”; an extension to all levels of a theorem of Hurewicz about 2ɸsets; a new proof of results of Kunugui, Novikov, Bourgain and the authors on Borel sets with sections of given class; extensions of results of Stern and Kechris. Our results are valid in arbitrary Polish spaces, and for the classes in Lavrentieffs and Wadge’s hierarchies.

For each countable ordinal $\xi$ and pair $(A_0, A_1)$ of disjoint analytic subsets of $2^\omega$, we define a closed game $J_\xi(A_0, A_1)$ and a complete $\Pi_\xi^O$ subset $H_\xi$ of $2^\omega$ such that (i) a winning strategy for player I constructs a $\sum_\xi^O$ set separating $A_0$ from $A_1$; and (ii) a winning strategy for player II constructs a continuous map $\varphi: 2^\omega \rightarrow A_0 \cup A_1$ with $\varphi^{-1}(A_0) = H_\xi$. Applications of this construction include: A proof in second order arithmetics of the statement "every $\Pi_\xi^O$ non $\sum_\xi^O$ set is $\Pi_\xi^O$-complete"; an extension to all levels of a theorem of Hurewicz about $\sum_2^O$ sets; a new proof of results of Kunugui, Novikov, Bourgain and the authors on Borel sets with sections of given class; extensions of results of Stern and Kechris. Our results are valid in arbitrary Polish spaces, and for the classes in Lavrentieff's and Wadge's hierarchies.

This chapter discusses linear orders in (ѡ)ѡ under eventual dominance. If (ѡ)ѡ is the set of functions f : ѡ →ѡ ordered by eventual dominance, then under this ordering, (ѡ)ѡ embeds every linear ordering of power < א1. It is proved that 2א0 > א1 and every linear ordering of cardinality <2א0 are embeddable in (ѡ)ѡ.

A subset of the Cantor space ω2 is called monotone iff it is closed upward under the partial ordering ≤ defined by x ≤ y iff x(n) ≤ y(n) for all n ∈ ω. A set is -positive (-positive) iff it is monotone and -positive set is a countable union of -positive sets; a -positive set is a countable intersection of -positive sets. (See Cenzer [2] for background information on these concepts.) It is clear that any -positive set is and monotone; the converse holds for n ≤ 2 [2] and was conjectured by Dyck to hold for greater n. In this note, we will disprove this conjecture by giving examples of monotone sets (for n ≥ 3) which are not even -positive.
First we note a few isomorphisms. The space (ω2, ≤) is isomorphic to the space (ω2 ≥), so instead of monotone and positive sets we may construct hereditary and negative sets (the analogous notions with “closed upward” replaced by “closed downward”). Also, (ω2, ≤) is isomorphic to ((ω), ⊆), where denotes the power set operator, or to ((S), ⊆) for any countably infinite set S.
In order to remove extraneous notation from the proofs, we state the results in an abstract form (whose generality is deceptive).

Kantorovich and Livenson [6] initiated the study of infinitary Boolean operations applied to the subsets of the Baire space and related spaces. It turns out that a number of interesting collections of subsets of the Baire space, such as the collection of Borel sets of a given type (e.g. the Fσ sets) or the collection of analytic sets, can be expressed as the range of an ω-ary Boolean operation applied to all possible ω-sequences of clopen sets. (Such collections are called clopen-ω-Boolean.) More recently, the ranges of I-ary Boolean operations for uncountable I have been considered; specific questions include whether the collection of Borel sets, or the collection of sets at finite levels in the Borel hierarchy, is clopen-I-Boolean.
The main purpose of this paper is to give a characterization of those collections of subsets of the Baire space (or similar spaces) that are clopen-I-Boolean for some I. The Baire space version can be stated as follows: a collection of subsets of the Baire space is clopen-I-Boolean for some I iff it is nonempty and closed downward and σ-directed upward under Wadge reducibility, and in this case we may take I = ω2. The basic method of proof is to use discrete subsets of spaces of the form K2 to put a number of smaller clopen-I-Boolean classes together to form a large one. The final section of the paper gives converse results indicating that, at least in some cases, ω2 cannot be replaced by a smaller index set.

Let A and B be subsets of the space 2
N
of sets of natural numbers. A is said to be Wadge reducible to B if there is a continuous map Φ from 2
N
into 2
N
such that A = Φ
−1 (B); A is said to be monotone reducible to B if in addition the map Φ is monotone, that is, a ⊂ b implies Φ(a) ⊂ Φ(b). The set A is said to be monotone if a ∈ A and a ⊂ b imply b ∈ A. For monotone sets, it is shown that, as for Wadge reducibility, sets low in the arithmetical hierarchy are nicely ordered. The sets are all reducible to the ( but not ) sets, which are in turn all reducible to the strictly sets, which are all in turn reducible to the strictly sets. In addition, the nontrivial sets all have the same degree for n ≤ 2. For Wadge reducibility, these results extend throughout the Borel hierarchy. In contrast, we give two natural strictly monotone sets which have different monotone degrees. We show that every monotone set is actually positive. We also consider reducibility for subsets of the space of compact subsets of 2
N
. This leads to the result that the finitely iterated Cantor-Bendixson derivative Dn
is a Borel map of class exactly 2n, which answers a question of Kuratowski.

Set theory, North-Holland

- K Kunen

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sse's order type conjecture

- On Frai

_,
On Frai'sse's order type conjecture, Ann. of Math. 93 (1971), 89-111.

Some problems in set theory and model theory, Doctoral Dissertation

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
11. A. Miller, Some problems in set theory and model theory, Doctoral Dissertation,
Univ. of California, Berkeley, 1978.

Linear orders xn (w)"' under eventual dominance, Logic Colloquium '78

- R Laver

R. Laver, Linear orders xn (w)"' under eventual dominance, Logic Colloquium '78 (M. BofTa et
al., eds.), North-Holland, Amsterdam, 1979.