Index

In the main items, Greek letters and mathematical symbols are alphabetized phoneti-

cally,disregardinghyphens,readingsuperscriptsbeforesubscripts,and(wherethefont

matters)givingprioritytolightfaceratherthanboldface: thusΣ2

1

isread sigmaonetwo,

it comes after Σn,

1

which is read sigmaonen and just before Σ2.

1

This convention is not

observedwithin eachitem, wheresub-andsubsub-itemsareordered“logically”, most

often following the order in which the terms occur in the text.

A , A

κ,52

◦ closure of the Lusin pointclasses under A ,

2B.5, 56

◦ closure of ¬Γ under A (Γ a Spector point-

class), 4C.6, 161

A-sets, same as analytic sets,

Σ1

1

absolutely Γ-inductive pointset, 312

absolutely measurable

◦ function, 84

◦ pointset, 83

- from determinacy hypotheses, 6A.18, 229

- for

Σ1

2

if κ → (ℵ1), 6G.12, 289

- for Σ2

1

if (∀α)[card(N ∩ L(α)) = ℵ0],

8G.9, 425

absoluteness

◦ basic theory, 8E

◦ of a notion, for a collection of classes, 391

◦ ZF-absoluteness, 392

◦ of a function as an operation, 399

◦ of a set as a condition, 399

◦ of Σ1

2

pointsets as conditions, 8F.9, 409

AC, Axiom of Choice, 374

◦ when it is needed, 7F

◦ in L, 8F.3, 403

AD, Axiom of (full) Determinacy, 292basic the-

ory: 7D

AD, Axiom of (full) Determinacy, 292

adequate pointclass, 119

ℵ1 is measurable, under AD, 7D.18, 338

Alg(M,J) (algebraic points over M,relative to

J), 423

α#,

see remarkable character

|α| (ordinal coded by α), 147

α(n) = α(0),...,α(n − 1) , 115

(α)i,31

α0,...,αk−1 , 31

ambiguous Borel pointclasses, see ∆n, 0 ∆0

analysis (the structure of), 356

analytic sets, same as

Σ1

1

analytical pointsets, 118

◦ of type 0,1, formally definable, 8B.3, 367,

8B.12, 369

◦ see also Kleene pointclasses

Approximation Theorem, 2H.1, 80

◦ over standard models of ZF, 8G.8, 423

arithmetic (structure of), 355

arithmetic subtraction k −

·

n,92

arithmetical pointsets, 118

◦ properly included in

∆1,

1

3F.8, 130

◦ formally defined, 8B.2, 367, 8B.11, 368

◦ see also Kleene pointclasses

A(u) (subgame of A at u), 219

Axiom

◦ of Choice, AC, see AC

◦ of Dependent Choices, DC, see DC

◦ of Determinacy, AD, see AD

◦ see also ZF,ZFC,ZFL

Baire Category Theorem, 2H.2, 82

Baire functions, 45

◦ relation with Borel functions, 1G.18, 45 –

1G.21, 46

◦ continuous on a comeager G , 2H.10, 84

◦ see also functions

Baire measurable functions, same as Baire func-

tions

Baire property, see property of Baire

Baire space, N,9

◦ homeomorphic with the irrational numbers,

9

◦continuouslysurjectedontoeveryPolishspace,

1A.1, 10, 1G.2, 38

◦ Borel isomorphic with every perfect product

space, 1G.4, 41

◦ recursively presented, 97

◦recursivelysurjectedontoeveryproductspace,

3D.14, 116, 3E.6, 121

◦

∆1-isomorphic

1

with every perfect product

space, 3E.7, 122

Baire-de la Vallee-Poussin class ≤ , 47

Banach-Mazur game

G∗∗(A),

X

226

◦ basic theory: 6A.13, 226 – 6A.16, 227

bar (backwards) induction and recursion, 62

basic nbhds (center, radius), 11

◦ recursively presented, 98

◦ absolutely presented, 401

basic space, 14

basis, 179

◦

∆1

1

not a basis for

Π0,

1

4D.10, 170

◦

forΣ1

2

(Novikov-Kondo-Addison), 4E.5, 179

491