The axiom of determinateness and reduction principles in the analytical hierarchy

Martin, Donald A.

doi:10.1090/s0002-9904-1968-11995-0

Cited by 138 publications

(69 citation statements)

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“…(A set S is degree invariant if ∀X ≡ T Y (X ∈ S ↔ Y ∈ S), and it is cofinal if ∀Z ∈ 2 ω ∃X T Z (X ∈ S).) Γ -Turing determinacy was introduced by Martin, who showed that it follows from plain Γ -determinacy [Mar68]. LEMMA 2.3 (ZF).…”

Section: The Proof Of the Main Theoremmentioning

confidence: 99%

Analytic Equivalence Relations Satisfying Hyperarithmetic-Is-Recursive

Montalbán

2015

Forum math. Sigma

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We prove, in ZF + Σ 1 2 -determinacy, that, for any analytic equivalence relation E, the following three statements are equivalent: (1) E does not have perfectly many classes, (2) E satisfies hyperarithmetic-is-recursive on a cone, and (3) relative to some oracle, for every equivalence class[Y ] E we have that a real X computes a member of the equivalence class if and only if ω1 . We also show that the implication from (1) to (2) is equivalent to the existence of sharps over Z F.

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Section: The Proof Of the Main Theoremmentioning

confidence: 99%

Analytic Equivalence Relations Satisfying Hyperarithmetic-Is-Recursive

Montalbán

2015

Forum math. Sigma

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“…The fundamental theorems connecting determinacy, norms and scales are the "Periodicity Theorems" of [AdMo68], [Mar68] and [Mo71]. In the following we shall need the first two Periodicity Theorems in special cases:…”

Section: Restricted Dual Laver Forcingmentioning

confidence: 99%

Techniques for approaching the dual Ramsey property in the projective hierarchy

Halbeisen¹,

Löwe²

2001

Pacific J. Math.

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“…Prewellordering (nj) can be proved by classical arguments and Prewellordering (Sj) was proved by Moschovakis; see [37]. Finally Martin [27] and independently Moschovakis [4] proved Prewellordering (n|"+1), Prewellordering (2^+2) for any n, assuming projective determinacy (PD) if n > 0. Here PD is the hypothesis that every projective set is determined.…”

Section: Xga -» A(x)mentioning

confidence: 99%

The theory of countable analytical sets

Kechris¹

1975

Trans. Amer. Math. Soc.

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ABSTRACT. 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 II set of reals, 2). The internal structure of the sets €" is then investigated in detail, the point of departure being the fact that 1 each <2n is a set of An-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, uj-models of analysis, higherlevel analogs of the constructible universe, inductive definability, etc.It is a classical theorem of effective descriptive set theory that a 2 J thin (i.e. containing no perfect set) subset of the continuum is countable and in fact contains only A{ reals. As a consequence, among the countable 2} sets of reals there is no largest one. Solovay [41] showed that every thin 2] set contains only constructible reals and therefore (in sharp contrast with the previous case), assuming a measurable cardinal exists, there is a largest countable 1,\ set of reals, namely the set of constructible ones. In [19] Moschovakis and the author extended Solovay's theorem to all even levels of the analytical hierarchy. It was proved there that, assuming projective determinacy (PD), there is a largest countable X\n set f°r all n> 1, which we denote by C2"-The first main result we prove in this paper (see §1) is the existence of a largest countable Il2" + 1 set (which we denote by C2w + i) for all n>0, assuming PD again. For w = 0 our proof shows, in ZF + DC only, the existence of a largest thin II} set of reals C,, a fact which was also independently discovered by D. Guaspari [12] and G. E. Sacks [38]. (To complete the picture, we remark here that it was shown in [17], using PD, that no largest countable 2L+i or n2n sets exist.) Once the existence of the largest countable sets C" is established the rest of § 1 is devoted to the study of their internal structure. We show here that C" is actually a set of A¿-degrees which is iwell ordered under the usual ordering of An-degrees.

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The axiom of determinateness and reduction principles in the analytical hierarchy

Cited by 138 publications

References 4 publications

Analytic Equivalence Relations Satisfying Hyperarithmetic-Is-Recursive

Analytic Equivalence Relations Satisfying Hyperarithmetic-Is-Recursive

Techniques for approaching the dual Ramsey property in the projective hierarchy

The theory of countable analytical sets

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