Universal sets for pointsets properly on the nth level of the projective hierarchy

2013 ◽  
Vol 78 (1) ◽  
pp. 237-244
Author(s):  
Greg Hjorth ◽  
Leigh Humphries ◽  
Arnold W. Miller
Keyword(s):  
Projective Hierarchy

AbstractThe Axiom of Projective Determinacy implies the existence of a universal set for every n ≥ 1. Assuming there exists a universal set. In ZFC there is a universal set for every α.

scholarly journals Interpreting the projective hierarchy in expansions of the real line

2014 ◽  
Vol 142 (9) ◽  
pp. 3259-3267 ◽  
Author(s):  
Philipp Hieronymi ◽  
Michael Tychonievich
Keyword(s):  
Real Line ◽  
The Real ◽  
Projective Hierarchy

scholarly journals Models of Set Theory in which Nonconstructible Reals First Appear at a Given Projective Level

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 910 ◽  
Author(s):  
Vladimir Kanovei ◽  
Vassily Lyubetsky
Keyword(s):  
Lower Limit ◽  
Set Theory ◽  
Generic Extension ◽  
Projective Class ◽  
Almost Disjoint ◽  
Projective Hierarchy ◽  
Models Of Set Theory

Models of set theory are defined, in which nonconstructible reals first appear on a given level of the projective hierarchy. Our main results are as follows. Suppose that n ≥ 2 . Then: 1. If it holds in the constructible universe L that a ⊆ ω and a ∉ Σ n 1 ∪ Π n 1 , then there is a generic extension of L in which a ∈ Δ n + 1 1 but still a ∉ Σ n 1 ∪ Π n 1 , and moreover, any set x ⊆ ω , x ∈ Σ n 1 , is constructible and Σ n 1 in L . 2. There exists a generic extension L in which it is true that there is a nonconstructible Δ n + 1 1 set a ⊆ ω , but all Σ n 1 sets x ⊆ ω are constructible and even Σ n 1 in L , and in addition, V = L [ a ] in the extension. 3. There exists an generic extension of L in which there is a nonconstructible Σ n + 1 1 set a ⊆ ω , but all Δ n + 1 1 sets x ⊆ ω are constructible and Δ n + 1 1 in L . Thus, nonconstructible reals (here subsets of ω ) can first appear at a given lightface projective class strictly higher than Σ 2 1 , in an appropriate generic extension of L . The lower limit Σ 2 1 is motivated by the Shoenfield absoluteness theorem, which implies that all Σ 2 1 sets a ⊆ ω are constructible. Our methods are based on almost-disjoint forcing. We add a sufficient number of generic reals to L , which are very similar at a given projective level n but discernible at the next level n + 1 .

The Projective Hierarchy

2017 ◽  
pp. 74-78
Author(s):  
Joseph R. Shoenfield
Keyword(s):  
Projective Hierarchy

Optimal Proofs of Determinacy

1995 ◽  
Vol 1 (3) ◽  
pp. 327-339 ◽  
Author(s):  
Itay Neeman
Keyword(s):  
Set Theory ◽  
Large Cardinals ◽  
The Other ◽  
Descriptive Set Theory ◽  
Transfer Theorem ◽  
Axiom Of Determinacy ◽  
Structural Connection ◽  
Projective Hierarchy ◽  
Descriptive Set

In this paper I shall present a method for proving determinacy from large cardinals which, in many cases, seems to yield optimal results. One of the main applications extends theorems of Martin, Steel and Woodin about determinacy within the projective hierarchy. The method can also be used to give a new proof of Woodin's theorem about determinacy in L(ℝ).The reason we look for optimal determinacy proofs is not only vanity. Such proofs serve to tighten the connection between large cardinals and descriptive set theory, letting us bring our knowledge of one subject to bear on the other, and thus increasing our understanding of both. A classic example of this is the Harrington-Martin proof that -determinacy implies -determinacy. This is an example of a transfer theorem, which assumes a certain determinacy hypothesis and proves a stronger one. While the statement of the theorem makes no mention of large cardinals, its proof goes through 0#, first proving that-determinacy ⇒ 0# exists,and then that0# exists ⇒ -determinacyMore recent examples of the connection between large cardinals and descriptive set theory include Steel's proof thatADL(ℝ) ⇒ HODL(ℝ) ⊨ GCH,see [9], and several results of Woodin about models of AD+, a strengthening of the axiom of determinacy AD which Woodin has introduced. These proofs not only use large cardinals, but also reveal a deep, structural connection between descriptive set theoretic notions and notions related to large cardinals.

Boolean operations, Borel sets, and Hausdorff's question

1996 ◽  
Vol 61 (4) ◽  
pp. 1287-1304
Author(s):  
Abhijit Dasgupta
Keyword(s):  
Polish Space ◽  
Descriptive Set Theory ◽  
Boolean Operation ◽  
Boolean Operations ◽  
Borel Sets ◽  
Systematic Fashion ◽  
Borel Hierarchy ◽  
The Difference ◽  
Projective Hierarchy ◽  
Descriptive Set

The study of infinitary Boolean operations was undertaken by the early researchers of descriptive set theory soon after Suslin's discovery of the important operation. The first attempt to lay down their theory in a systematic fashion was the work of Kantorovich and Livenson [5], where they call these the analytical operations. Earlier, Hausdorff had introduced the δs operations — essentially same as the monotoneω-ary Boolean operations, and Kolmogorov, independently of Hausdorff, had discovered the same objects, which were used in his study of the R operator.The ω-ary Boolean operations turned out to be closely related to most of the classical hierarchies over a fixed Polish space X, including, e. g., the Borel hierarchy (), the difference hierarchies of Hausdorff (Dη()), the C-hierarchy (Cξ) of Selivanovski, and the projective hierarchy (): for each of these hierarchies, every level can be expressed as the range of an ω-ary Boolean operation applied to all possible sequences of open subsets of X. In the terminology of Dougherty [3], every level is “open-ω-Boolean” (if and are collections of subsets of X and I is any set, is said to be -I-Boolean if there exists an I-ary Boolean operation Φ such that = Φ, i. e. is the range of Φ restricted to all possible I-sequences of sets from ). If in addition, the space X has a basis consisting of clopen sets, then the levels of the above hierarchies are also “clopen-ω-Boolean.”

Countable functionals and the projective hierarchy

1981 ◽  
Vol 46 (2) ◽  
pp. 209-215 ◽  
Author(s):  
Dag Normann
Keyword(s):  
Equivalence Classes ◽  
Type Structure ◽  
Continuous Functional ◽  
Image Position ◽  
Projective Hierarchy ◽  
Continuous Functionals ◽  
Original Approach ◽  
Computable Functional

Kleene [7] and Kreisel [8] defined independently the countable (continuous) functionals. Kleene [7] defined the countable functionals of type k to be total functionals of type k acting in a continuous way when restricted to countable arguments of type k − 1. He also defined the associates for countable functionals. They are functions α: N → N containing information about how the functional acts on countable arguments. Kleene [7] showed that the countable functionals are closed under the computations derived from S1–S9 of his paper [6], and that every computable functional has a recursive associate.Kreisel defined the continuous functionals to be equivalence-classes of associates. By his definition it is meaningless to let a continuous functional act upon anything but continuous arguments.One disadvantage of Kleene's approach is that two different functionals may have the same associates We will later see that there may be two functionals φ1 and φ2 with the same associates but such that the relationsare not the same.In more recent papers on the countable functionals it is normal to regard the hierarchy 〈Ct(k)〉kϵN of countable functionals as a type-structure such that the functionals in Ct(k + 1) are maps from Ct(k) to N(Ct(0) = N), see e.g. Bergstra [1] and Gandy and Hyland [3].We will then enjoy the streamlined formalism of a type-structure in which S1–S9 have meaning, but avoid the ambiguities of Kleene's original approach.We will presuppose a brief familiarity with the theory of the countable functionals.

sets and singletons

1999 ◽  
Vol 64 (2) ◽  
pp. 590-616
Author(s):  
Kai Hauser ◽  
W. Hugh Woodin
Keyword(s):  
Core Model ◽  
Real Numbers ◽  
The Third ◽  
Sacks Forcing ◽  
Projective Hierarchy ◽  
Definable Elements

AbstractWe extend work of H. Friedman, L. Harrington and P. Welch to the third level of the projective hierarchy. Our main theorems say that (under appropriate background assumptions) the possibility to select definable elements of non-empty sets of reals at the third level of the projective hierarchy is equivalent to the disjunction of determinacy of games at the second level of the projective hierarchy and the existence of a core model (corresponding to this fragment of determinacy) which must then contain all real numbers. The proofs use Sacks forcing with perfect trees and core model techniques.

Nonuniformization results for the projective hierarchy

1991 ◽  
Vol 56 (2) ◽  
pp. 742-748 ◽  
Author(s):  
Steve Jackson ◽  
R. Daniel Mauldin
Keyword(s):  
Polish Spaces ◽  
Projective Hierarchy

AbstractLet X and Y be uncountable Polish spaces. We show in ZF that there is a coanalytic subset P of X × Y with countable sections which cannot be expressed as the union of countably many partial coanalytic, or even PCA = , graphs. If X = Y = ωω, P may be taken to be . Assuming stronger set theoretic axioms, we identify the least pointclass such that any such coanalytic P can be expressed as the union of countably many graphs in this pointclass. This last result is extended (under suitable hypotheses) to all levels of the projective hierarchy.

scholarly journals Polarized partitions on the second level of the projective hierarchy

2012 ◽  
Vol 163 (9) ◽  
pp. 1345-1357 ◽  
Author(s):  
Jörg Brendle ◽  
Yurii Khomskii
Keyword(s):  
Projective Hierarchy
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