DeterminacyFrom Wikipedia, the free encyclopedia

Contents

1 AD+ 11.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Axiom of determinacy 22.1 Types of game that are determined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Incompatibility of the axiom of determinacy with the axiom of choice . . . . . . . . . . . . . . . . 22.3 Infinite logic and the axiom of determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 Large cardinals and the axiom of determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Axiom of projective determinacy 53.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4 Axiom of real determinacy 6

5 Banach–Mazur game 75.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75.2 A simple proof: winning strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

6 Borel determinacy theorem 96.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

6.1.1 Gale–Stewart games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96.1.2 Winning strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96.1.3 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

6.2 Previous results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106.3 Borel determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106.4 Set-theoretic aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116.5 Stronger forms of determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

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7 Determinacy 137.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

7.1.1 Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137.1.2 Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137.1.3 Winning strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.1.4 Determined games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

7.2 Determinacy from elementary considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.3 Determinacy from ZFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.4 Determinacy and large cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7.4.1 Measurable cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.4.2 Woodin cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.4.3 Projective determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.4.4 Axiom of determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

7.5 Consequences of determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.5.1 Regularity properties for sets of reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.5.2 Periodicity theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.5.3 Applications to decidability of certain second-order theories . . . . . . . . . . . . . . . . . 177.5.4 Wadge determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

7.6 More general games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.6.1 Games in which the objects played are not natural numbers . . . . . . . . . . . . . . . . . 177.6.2 Games played on trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.6.3 Long games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.6.4 Games of imperfect information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7.7 Quasistrategies and quasideterminacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.8 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

8 Homogeneous tree 208.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

9 Homogeneously Suslin set 219.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

10 L(R) 2210.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2210.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2210.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2210.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

11 Lightface analytic game 24

12 Martin measure 25

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12.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2512.2 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2512.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

13 Measurable cardinal 2613.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2613.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2613.3 Real-valued measurable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2713.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2713.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2713.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

14 Property of Baire 2914.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2914.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2914.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

15 Rank-into-rank 3015.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

16 Tree (descriptive set theory) 3216.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

16.1.1 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3216.1.2 Branches and bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3216.1.3 Terminal nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

16.2 Relation to other types of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3216.3 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3316.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3316.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

17 Universally measurable set 3417.1 Finiteness condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3417.2 Example contrasting with Lebesgue measurability . . . . . . . . . . . . . . . . . . . . . . . . . . 3417.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

18 Woodin cardinal 3618.1 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3618.2 Hyper-Woodin cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3718.3 Weakly hyper-Woodin cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3718.4 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3718.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

19 Zero sharp 3819.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

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19.2 Statements that imply the existence of 0# . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3819.3 Statements equivalent to existence of 0# . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3919.4 Consequences of existence and non-existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3919.5 Other sharps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3919.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3919.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

20 Θ (set theory) 4120.1 Proof of existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4120.2 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 42

20.2.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4220.2.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4320.2.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Chapter 1

AD+

In set theory,AD+ is an extension, proposed byW. HughWoodin, to the axiom of determinacy. The axiom, which isto be understood in the context of ZF plus DCR (the axiom of dependent choice for real numbers), states two things:

1. Every set of reals is ∞-Borel.

2. For any ordinal λ less than Θ, any subset A of ωω, and any continuous function π:λω→ωω, the preimage π−1[A]is determined. (Here λω is to be given the product topology, starting with the discrete topology on λ.)

The second clause by itself is referred to as ordinal determinacy.

1.1 See also• Suslin’s problem

1.2 References• W.H. Woodin The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal (1999 Walter deGruyter) p. 618

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Chapter 2

Axiom of determinacy

In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by JanMycielski and Hugo Steinhaus in 1962. It refers to certain two-person games of length ω with perfect information.AD states that every such game in which both players choose natural numbers is determined; that is, one of the twoplayers has a winning strategy.The axiom of determinacy is inconsistent with the axiom of choice (AC); the axiom of determinacy implies thatall subsets of the real numbers are Lebesgue measurable, have the property of Baire, and the perfect set property.The last implies a weak form of the continuum hypothesis (namely, that every uncountable set of reals has the samecardinality as the full set of reals).Furthermore, AD implies the consistency of Zermelo–Fraenkel set theory (ZF). Hence, as a consequence of theincompleteness theorems, it is not possible to prove the relative consistency of ZF + AD with respect to ZF.

2.1 Types of game that are determined

Not all games require the axiom of determinacy to prove them determined. Games whose winning sets are closed aredetermined. These correspond to many naturally defined infinite games. It was shown in 1975 by Donald A. Martinthat games whose winning set is a Borel set are determined. It follows from the existence of sufficient large cardinalsthat all games with winning set a projective set are determined (see Projective determinacy), and that AD holds inL(R).

2.2 Incompatibility of the axiom of determinacy with the axiom of choice

The set S1 of all first player strategies in an ω-game G has the same cardinality as the continuum. The same is trueof the set S2 of all second player strategies. We note that the cardinality of the set SG of all sequences possible in Gis also the continuum. Let A be the subset of SG of all sequences which make the first player win. With the axiomof choice we can well order the continuum; furthermore, we can do so in such a way that any proper initial portiondoes not have the cardinality of the continuum. We create a counterexample by transfinite induction on the set ofstrategies under this well ordering:We start with the set A undefined. Let T be the “time” whose axis has length continuum. We need to consider allstrategies {s1(T)} of the first player and all strategies {s2(T)} of the second player to make sure that for every strategythere is a strategy of the other player that wins against it. For every strategy of the player considered we will generatea sequence which gives the other player a win. Let t be the time whose axis has length ℵ0 and which is used duringeach game sequence.

1. Consider the current strategy {s1(T)} of the first player.

2. Go through the entire game, generating (together with the first player’s strategy s1(T)) a sequence {a(1), b(2),a(3), b(4),...,a(t), b(t+1),...}.

3. Decide that this sequence does not belong to A, i.e. s1(T) lost.

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2.3. INFINITE LOGIC AND THE AXIOM OF DETERMINACY 3

4. Consider the strategy {s2(T)} of the second player.

5. Go through the next entire game, generating (together with the second player’s strategy s2(T)) a sequence{c(1), d(2), c(3), d(4),...,c(t), d(t+1),...}, making sure that this sequence is different from {a(1), b(2), a(3),b(4),...,a(t), b(t+1),...}.

6. Decide that this sequence belongs to A, i.e. s2(T) lost.

7. Keep repeating with further strategies if there are any, making sure that sequences already considered do notbecome generated again. (We start from the set of all sequences and each time we generate a sequence andrefute a strategy we project the generated sequence onto first player moves and onto second player moves, andwe take away the two resulting sequences from our set of sequences.)

8. For all sequences that did not come up in the above consideration arbitrarily decide whether they belong to A,or to the complement of A.

Once this has been done we have a game G. If you give me a strategy s1 then we considered that strategy at sometime T = T(s1). At time T, we decided an outcome of s1 that would be a loss of s1. Hence this strategy fails. Butthis is true for an arbitrary strategy; hence the axiom of determinacy and the axiom of choice are incompatible.

2.3 Infinite logic and the axiom of determinacy

Many different versions of infinitary logic were proposed in the late 20th century. One reason that has been given forbelieving in the axiom of determinacy is that it can be written as follows (in a version of infinite logic):∀G ⊆ Seq(S) :

∀a ∈ S : ∃a′ ∈ S : ∀b ∈ S : ∃b′ ∈ S : ∀c ∈ S : ∃c′ ∈ S... : (a, a′, b, b′, c, c′...) ∈ G OR∃a ∈ S : ∀a′ ∈ S : ∃b ∈ S : ∀b′ ∈ S : ∃c ∈ S : ∀c′ ∈ S... : (a, a′, b, b′, c, c′...) /∈ G

Note: Seq(S) is the set of all ω -sequences of S. The sentences here are infinitely long with a countably infinite list ofquantifiers where the ellipses appear.In an infinitary logic, this principle is therefore a natural generalization of the usual (de Morgan) rule for quantifiersthat are true for finite formulas, such as ∀a : ∃b : ∀c : ∃d : R(a, b, c, d) OR ∃a : ∀b : ∃c : ∀d : ¬R(a, b, c, d) .

2.4 Large cardinals and the axiom of determinacy

The consistency of the axiom of determinacy is closely related to the question of the consistency of large cardinalaxioms. By a theorem of Woodin, the consistency of Zermelo–Fraenkel set theory without choice (ZF) together withthe axiom of determinacy is equivalent to the consistency of Zermelo–Fraenkel set theory with choice (ZFC) togetherwith the existence of infinitely many Woodin cardinals. Since Woodin cardinals are strongly inaccessible, if AD isconsistent, then so are an infinity of inaccessible cardinals.Moreover, if to the hypothesis of an infinite set of Woodin cardinals is added the existence of a measurable cardinallarger than all of them, a very strong theory of Lebesgue measurable sets of reals emerges, as it is then provable thatthe axiom of determinacy is true in L(R), and therefore that every set of real numbers in L(R) is determined.

2.5 See also

• Axiom of real determinacy (ADR)

• AD+, a variant of the axiom of determinacy formulated by Woodin

• Axiom of quasi-determinacy (ADQ)

• Martin measure

4 CHAPTER 2. AXIOM OF DETERMINACY

2.6 References• Jech, Thomas (2002). Set theory, third millennium edition (revised and expanded). Springer. ISBN 3-540-44085-2.

• Kanamori, Akihiro (2000). The Higher Infinite (2nd ed.). Springer. ISBN 3-540-00384-3.

• Martin, Donald A.; Steel, John R. (Jan 1989). “A Proof of Projective Determinacy”. Journal of the AmericanMathematical Society 2 (1): 71–125. doi:10.2307/1990913. JSTOR 1990913.

• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.

• Mycielski, Jan; Steinhaus, H. (1962). “A mathematical axiom contradicting the axiom of choice”. Bulletinde l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 10: 1–3.ISSN 0001-4117. MR 0140430.

• Woodin,W.Hugh (1988). “Supercompact cardinals, sets of reals, andweakly homogeneous trees”. Proceedingsof the National Academy of Sciences of theUnited States of America 85 (18): 6587–6591. doi:10.1073/pnas.85.18.6587.PMC 282022. PMID 16593979.

2.7 Further reading• Philipp Rohde, On Extensions of the Axiom of Determinacy, Thesis, Department of Mathematics, Universityof Bonn, Germany, 2001

• Telgársky, R.J. Topological Games: On the 50th Anniversary of the Banach-Mazur Game, Rocky Mountain J.Math. 17 (1987), pp. 227–276. (3.19 MB)

Chapter 3

Axiom of projective determinacy

In mathematical logic, projective determinacy is the special case of the axiom of determinacy applying only toprojective sets.The axiom of projective determinacy, abbreviated PD, states that for any two-player game of perfect informationof length ω in which the players play natural numbers, if the victory set (for either player, since the projective setsare closed under complementation) is projective, then one player or the other has a winning strategy.The axiom is not a theorem of ZFC (assuming ZFC is consistent), but unlike the full axiom of determinacy (AD),which contradicts the axiom of choice, it is not known to be inconsistent with ZFC. PD follows from certain largecardinal axioms, such as the existence of infinitely many Woodin cardinals.PD implies that all projective sets are Lebesgue measurable (in fact, universally measurable) and have the perfectset property and the property of Baire. It also implies that every projective binary relation may be uniformized by aprojective set.

3.1 References• Martin, Donald A. and John R. Steel (Jan 1989). “A Proof of Projective Determinacy”. Journal of the Amer-ican Mathematical Society (American Mathematical Society) 2 (1): 71–125. doi:10.2307/1990913. JSTOR1990913.

• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.

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Chapter 4

Axiom of real determinacy

In mathematics, the axiom of real determinacy (abbreviated as ADR) is an axiom in set theory. It states thefollowing:

Consider infinite two-person games with perfect information. Then, every game of length ω where bothplayers choose real numbers is determined, i.e., one of the two players has a winning strategy.

The axiom of real determinacy is a stronger version of the axiom of determinacy, which makes the same statementabout games where both players choose integers; it is inconsistent with the axiom of choice. ADR also implies theexistence of inner models with certain large cardinals.ADR is equivalent to AD plus the axiom of uniformization.

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Chapter 5

Banach–Mazur game

In general topology, set theory and game theory, aBanach–Mazur game is a topological game played by two players,trying to pin down elements in a set (space). The concept of a Banach–Mazur game is closely related to the concept ofBaire spaces. This game was the first infinite positional game of perfect information to be studied. It was introducedby Mazur as problem 43 in the Scottish book, and Mazur’s questions about it were answered by Banach.

5.1 Definition and properties

In what follows we will make use of the formalism defined in Topological game. A general Banach–Mazur game isdefined as follows: we have a topological space Y , a fixed subset X ⊂ Y , and a family W of subsets of Y thatsatisfy the following properties.

• Each member ofW has non-empty interior.

• Each non-empty open subset of Y contains a member ofW .

We will call this gameMB(X,Y,W ) . Two players, P1 and P2 , choose alternatively elementsW0 ,W1 , · · · ofWsuch thatW0 ⊃ W1 ⊃ · · · . The player P1 wins if and only ifX ∩ (∩n<ωWn) ̸= ∅ .The following properties hold.

• P2 ↑ MB(X,Y,W ) if and only if X is of the first category in Y (a set is of the first category or meagre if itis the countable union of nowhere-dense sets).

• Assuming that Y is a complete metric space, P1 ↑ MS(X,Y,W ) if and only if X is comeager in somenonempty open subset of Y .

• If X has the Baire property in Y , thenMB(X,Y,W ) is determined.

• Any winning strategy of P2 can be reduced to a stationary winning strategy.

• The siftable and strongly-siftable spaces introduced by Choquet can be defined in terms of stationary strategiesin suitable modifications of the game. Let BM(X) denote a modification ofMB(X,Y,W ) where X = Y ,W is the family of all nonempty open sets inX , andP2 wins a play (W0,W1, · · · ) if and only if∩n<ωWn ̸= ∅. Then X is siftable if and only if P2 has a stationary winning strategy in BM(X) .

• A Markov winning strategy for P2 in BM(X) can be reduced to a stationary winning strategy. Furthermore,if P2 has a winning strategy in BM(X) , then she has a winning strategy depending only on two precedingmoves. It is still an unsettled question whether a winning strategy for P2 can be reduced to a winning strategythat depends only on the last two moves of P1 .

• X is called weakly α -favorable if P2 has a winning strategy in BM(X) . Then, X is a Baire space if andonly if P1 has no winning strategy inBM(X) . It follows that each weakly α -favorable space is a Baire space.

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8 CHAPTER 5. BANACH–MAZUR GAME

Many other modifications and specializations of the basic game have been proposed: for a thorough account of these,refer to [1987]. The most common special case, calledMB(X, J) , consists in letting Y = J , i.e. the unit interval[0, 1] , and in letting W consist of all closed intervals [a, b] contained in [0, 1] . The players choose alternativelysubintervals J0, J1, · · · of J such that J0 ⊃ J1 ⊃ · · · , and P1 wins if and only if X ∩ (∩n<ωJn) ̸= ∅ . P2 wins ifand only if X ∩ (∩n<ωJn) = ∅ .

5.2 A simple proof: winning strategies

It is natural to ask for what setsX does P2 have a winning strategy. Clearly, ifX is empty, P2 has a winning strategy,therefore the question can be informally rephrased as how “small” (respectively, “big”) does X (respectively, thecomplement ofX in Y ) have to be to ensure that P2 has a winning strategy. To give a flavor of how the proofs usedto derive the properties in the previous section work, let us show the following fact.Fact: P2 has a winning strategy if X is countable, Y is T1, and Y has no isolated points.

Proof: Let the elements ofX be x1, x2, · · · . Suppose thatW1 has been chosen byP1 , and letU1 be the (non-empty)interior of W1 . Then U1 \ {x1} is a non-empty open set in Y , so P2 can choose a member W2 of W containedin this set. Then P1 chooses a subset W3 of W2 and, in a similar fashion, P2 can choose a member W4 ⊂ W3 thatexcludes x2 . Continuing in this way, each point xn will be excluded by the set W2n , so that the intersection of alltheWn will have empty intersection with X . Q.E.DThe assumptions on Y are key to the proof: for instance, if Y = {a, b, c} is equipped with the discrete topology andW consists of all non-empty subsets of Y , then P2 has no winning strategy if X = {a} (as a matter of fact, heropponent has a winning strategy). Similar effects happen if Y is equipped with indiscrete topology andW = {Y } .A stronger result relatesX to first-order sets.Fact: Let Y be a topological space, letW be a family of subsets of Y satisfying the two properties above, and letXbe any subset of Y . P2 has a winning strategy if and only if X is meagre.This does not imply that P1 has a winning strategy ifX is not meagre. In fact, P1 has a winning strategy if and onlyif there is some Wi ∈ W such that X ∩ Wi is a comeagre subset of Wi . It may be the case that neither playerhas a winning strategy: when Y is [0, 1] andW consists of the closed intervals [a, b] , the game is determined if thetarget set has the property of Baire, i.e. if it differs from an open set by a meagre set (but the converse is not true).Assuming the axiom of choice, there are subsets of [0, 1] for which the Banach–Mazur game is not determined.

5.3 References• [1957] Oxtoby, J.C. The Banach–Mazur game and Banach category theorem, Contribution to the Theory ofGames, Volume III, Annals of Mathematical Studies 39 (1957), Princeton, 159–163

• [1987] Telgársky, R. J. Topological Games: On the 50th Anniversary of the Banach–Mazur Game, RockyMountain J. Math. 17 (1987), pp. 227–276. (3.19 MB)

• [2003] Julian P. Revalski The Banach–Mazur game: History and recent developments, Seminar notes, Pointe-a-Pitre, Guadeloupe, France, 2003–2004

Chapter 6

Borel determinacy theorem

In descriptive set theory, the Borel determinacy theorem states that any Gale-Stewart game whose payoff set is aBorel set is determined, meaning that one of the two players will have a winning strategy for the game. It was provedby Donald A. Martin in 1975. The theorem is applied in descriptive set theory to show that Borel sets in Polish spaceshave regularity properties such as the perfect set property and the property of Baire.The theorem is also known for its metamathematical properties. In 1971, before the theorem was proved, HarveyFriedman showed that any proof of the theorem in Zermelo-Fraenkel set theory must make repeated use of the axiomof replacement. Later results showed that stronger determinacy theorems cannot be proven in Zermelo-Fraenkel settheory, although they are relatively consistent with it if certain large cardinals are consistent.

6.1 Background

6.1.1 Gale–Stewart games

Main article: Determinacy

A Gale–Stewart game is a two-player game of perfect information. The game is defined using a set A, and isdenoted GA. The two players alternate turns, and each player is aware of all moves before making the next one. Oneach turn, each player chooses a single element of A to play. The same element may be chosen more than oncewithout restriction. The game can be visualized through the following diagram, in which the moves are made fromleft to right, with the moves of player I above and the moves of player II below.

I a1 a3 a5 · · ·II a2 a4 a6 · · ·

The play continues without end, so that a single play of the game determines an infinite sequence ⟨a1, a2, a3 . . .⟩ ofelements of A. The set of all such sequences is denoted Aω. The players are aware, from the beginning of the game,of a fixed payoff set (a.k.a. winning set) that will determine who wins. The payoff set is a subset of Aω. If the infinitesequence created by a play of the game is in the payoff set, then player I wins. Otherwise, player II wins; there areno ties.

6.1.2 Winning strategies

A winning strategy for a player is a function that tells the player what move to make from any position in the game,such that if the player follows the function he or she will surely win. More specifically, a winning strategy for playerI is a function f that takes as input sequences of elements of A of even length and returns an element of A, such thatplayer I will win every play of the form

I a1 = f(⟨⟩) a3 = f(⟨a1, a2⟩) a5 = f(⟨a1, a2, a3, a4⟩) · · ·II a2 a4 a6 · · · .

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10 CHAPTER 6. BOREL DETERMINACY THEOREM

A winning strategy for player II is a function g that takes odd-length sequences of elements of A and returns elementsof A, such that player II will win every play of the form

I a1 a3 a5 · · ·II a2 = g(⟨a1⟩) a4 = g(⟨a1, a2, a3⟩) a6 = g(⟨a1, a2, a3, a4, a5⟩) · · · .

At most one player can have a winning strategy; if both players had winning strategies, and played the strategiesagainst each other, only one of the two strategies could win that play of the game. If one of the players has a winningstrategy for a particular payoff set, that payoff set is said to be determined.

6.1.3 Topology

For a given set A, whether a subset of Aω will be determined depends to some extent on its topological structure.For the purposes of Gale–Stewart games, the set A is endowed with the discrete topology, and Aω endowed withthe resulting product topology, where Aω is viewed as a countably infinite topological product of A with itself. Inparticular, when A is the set {0,1}, the topology defined on Aω is exactly the ordinary topology on Cantor space, andwhen A is the set of natural numbers, it is the ordinary topology on Baire space.The set Aω can be viewed as the set of paths through a certain tree, which leads to a second characterization of itstopology. The tree consists of all finite sequences of elements of A, and the children of a particular node σ of the treeare exactly the sequences that extend σ by one element. Thus if A = { 0, 1 }, the first level of the tree consists of thesequences ⟨ 0 ⟩ and ⟨ 1 ⟩; the second level consists of the four sequences ⟨ 0, 0 ⟩, ⟨ 0, 1 ⟩, ⟨ 1, 0 ⟩, ⟨ 1, 1 ⟩; and so on.For each of the finite sequences σ in the tree, the set of all elements of Aω that begin with σ is a basic open set in thetopology on A. The open sets of Aω are precisely the sets expressible as unions of these basic open sets. The closedsets, as usual, are those whose complement is open.The Borel sets ofAω are the smallest class of subsets ofAω that includes the open sets and is closed under complementand countable union. That is, the Borel sets are the smallest σ-algebra of subsets of Aω containing all the open sets.The Borel sets are classified in the Borel hierarchy based on how many times the operations of complement andcountable union are required to produce them from open sets.

6.2 Previous results

Gale and Stewart (1953) proved that if the payoff set is an open or closed subset of Aω then the Gale–Stewart gamewith that payoff set is always determined. Over the next twenty years, this was extended to slightly higher levelsof the Borel hierarchy through ever more complicated proofs. This led to the question of whether the game mustbe determined whenever the payoff set is a Borel subset of Aω. It was known that, using the axiom of choice, it ispossible to construct a subset of {0,1}ω that is not determined (Kechris 1995, p. 139).Harvey Friedman (1971) proved that that any proof that all Borel subsets of Cantor space ({0,1}ω ) were determinedwould require repeated use of the axiom of replacement, an axiom not typically required to prove theorems about“small” objects such as Cantor space.

6.3 Borel determinacy

Donald A. Martin (1975) proved that for any setA, all Borel subsets of Aω are determined. Because the original proofwas quite complicated, Martin published a shorter proof in 1982 that did not require as much technical machinery.In his review of Martin’s paper, Drake describes the second proof as “surprisingly straightforward.”The field of descriptive set theory studies properties of Polish spaces (essentially, complete separable metric spaces).The Borel determinacy theorem has been used to establish many properties of Borel subsets of these spaces. Forexample, all Borel subsets of Polish spaces have the perfect set property and the property of Baire.

6.4. SET-THEORETIC ASPECTS 11

6.4 Set-theoretic aspects

The Borel determinacy theorem is of interest for its metamethematical properties as well as its consequences indescriptive set theory.Determinacy of closed sets of Aω for arbitrary A is equivalent to the axiom of choice over ZF (Kechris 1995, p.139). When working in set-theoretical systems where the axiom of choice is not assumed, this can be circumventedby considering generalized strategies known as quasistrategies (Kechris 1995, p. 139) or by only considering gameswhere A is the set of natural numbers, as in the axiom of determinacy.Zermelo set theory (Z) is Zermelo-Fraenkel set theory without the axiom of replacement. It differs from ZF in thatZ does not prove that the powerset operation can be iterated uncountably many times beginning with an arbitrary set.In particular, Vω ₊ ω, a particular countable level of the cumulative hierarchy, is a model of Zermelo set theory. Theaxiom of replacement, on the other hand, is only satisfied by Vκ for significantly larger values of κ, such as when κis a strongly inaccessible cardinal. Friedman’s theorem of 1971 showed that there is a model of Zermelo set theory(with the axiom of choice) in which Borel determinacy fails, and thus Zermelo set theory cannot prove the Boreldeterminacy theorem.

6.5 Stronger forms of determinacy

Main article: determinacy

Several set-theoretic principles about determinacy stronger than Borel determinacy are studied in descriptive settheory. They are closely related to large cardinal axioms.The axiom of projective determinacy states that all projective subsets of a Polish space are determined. It is knownto be unprovable in ZFC but relatively consistent with it and implied by certain large cardinal axioms. The existenceof a measurable cardinal is enough to imply over ZFC that all analytic subsets of Polish spaces are determined.The axiom of determinacy states that all subsets of all Polish spaces are determined. It is inconsistent with ZFC butin ZF + DC (Zermelo-Fraenkel set theory plus the axiom of dependent choice) it is equiconsistent with certain largecardinal axioms.

6.6 References

• Friedman, Harvey (1971). “Higher set theory and mathematical practice”. Annals of Mathematical Logic 2(3): 325–357. doi:10.1016/0003-4843(71)90018-0.

• L. Bukovský, reviewer, Mathematical Reviews, MR 284327.

• Gale, D. and F. M. Stewart (1953). “Infinite games with perfect information”. Contributions to the theory ofgames, vol. 2. Annals of Mathematical Studies, vol. 28 28. Princeton University Press. pp. 245–266.

• S. Sherman, reviewer, Mathematical Reviews, MR 54922.

• Alexander Kechris (1995). Classical descriptive set theory. Graduate texts in mathematics 156. ISBN 0-387-94374-9.

• Martin, Donald A. (1975). “Borel determinacy”. Annals of Mathematics. Second Series 102 (2): 363–371.doi:10.2307/1971035.

• John Burgess, reviewer. Mathematical Reviews, MR 403976.

• Martin, Donald A. (1982). “A purely inductive proof of Borel determinacy”. Recursion theory. Proc. Sympos.Pure Math (Proceedings of the AMS–ASL summer institute held in Ithaca, New York ed.). pp. 303–308.

• F. R. Drake, reviewer, Mathematical Reviews, MR 791065.

12 CHAPTER 6. BOREL DETERMINACY THEOREM

6.7 External links• Borel determinacy and metamathematics. Ross Bryant. Master’s thesis, University of North Texas, 2001.

Chapter 7

Determinacy

“Determined” redirects here. For the 2005 heavy metal song, see Determined (song).For other uses of “indeterminacy”, see Indeterminacy.

In set theory, a branch of mathematics, determinacy is the study of under what circumstances one or the other playerof a game must have a winning strategy, and the consequences of the existence of such strategies.

7.1 Basic notions

7.1.1 Games

The first sort of game we shall consider is the two-player game of perfect information of length ω, in which theplayers play natural numbers.In this sort of game we consider two players, often named I and II, who take turns playing natural numbers, withI going first. They play “forever"; that is, their plays are indexed by the natural numbers. When they're finished, apredetermined condition decides which player won. This condition need not be specified by any definable rule; it maysimply be an arbitrary (infinitely long) lookup table saying who has won given a particular sequence of plays.More formally, consider a subsetA of Baire space; recall that the latter consists of all ω-sequences of natural numbers.Then in the game GA, I plays a natural number a0, then II plays a1, then I plays a2, and so on. Then I wins the gameif and only if

⟨a0, a1, a2, . . .⟩ ∈ A

and otherwise II wins. A is then called the payoff set of GA.It is assumed that each player can see all moves preceding each of his moves, and also knows the winning condition.

7.1.2 Strategies

Informally, a strategy for a player is a way of playing in which his plays are entirely determined by the foregoingplays. Again, such a “way” does not have to be capable of being captured by any explicable “rule”, but may simplybe a lookup table.More formally, a strategy for player I (for a game in the sense of the preceding subsection) is a function that acceptsas an argument any finite sequence of natural numbers, of even length, and returns a natural number. If σ is such astrategy and <a0,…,a₂ -₁> is a sequence of plays, then σ(<a0,…,a₂ -₁>) is the next play I will make, if he is followingthe strategy σ. Strategies for II are just the same, substituting “odd” for “even”.Note that we have said nothing, as yet, about whether a strategy is in any way good. A strategy might direct a playerto make aggressively bad moves, and it would still be a strategy. In fact it is not necessary even to know the winningcondition for a game, to know what strategies exist for the game.

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14 CHAPTER 7. DETERMINACY

7.1.3 Winning strategies

A strategy iswinning if the player following it must necessarily win, no matter what his opponent plays. For exampleif σ is a strategy for I, then σ is a winning strategy for I in the game GA if, for any sequence of natural numbers tobe played by II, say <a1,a3,a5,…>, the sequence of plays produced by σ when II plays thus, namely

⟨σ(⟨⟩), a1, σ(⟨σ(⟨⟩), a1⟩), a3, . . .⟩

is an element of A.

7.1.4 Determined games

A (class of) game(s) is determined if for all instances of the game there is a winning strategy for one of the players(not necessarily the same player for each instance). Note that there cannot be a winning strategy for both players forthe same game, for if there were, the two strategies could be played against each other. The resulting outcome wouldthen, by hypothesis, be a win for both players, which is impossible.

7.2 Determinacy from elementary considerations

All finite games of perfect information in which draws do not occur are determined.Familiar real-world games of perfect information, such as chess or tic-tac-toe, are always finished in a finite numberof moves. If such a game is modified so that a particular player wins under any condition where the game wouldhave been called a draw, then it is always determined. The condition that the game is always over (i.e. all possibleextensions of the finite position result in a win for the same player) in a finite number of moves corresponds to thetopological condition that the set A giving the winning condition for GA is clopen in the topology of Baire space.For example, modifying the rules of chess to make drawn games a win for Black makes chess a determined game. Asit happens, chess has a finite number of positions and a draw-by-repetition rules, so with these modified rules, if playcontinues long enough without White having won, then Black can eventually force a win (due to the modification ofdraw = win for black).It is an instructive exercise to figure out how to represent such games as games in the context of this article.The proof that such games are determined is rather simple: Player I simply plays not to lose; that is, he plays to makesure that player II does not have a winning strategy after I's move. If player I cannot do this, then it means playerII had a winning strategy from the beginning. On the other hand, if player I can play in this way, then he must win,because the game will be over after some finite number of moves, and he can't have lost at that point.This proof does not actually require that the game always be over in a finite number of moves, only that it be over ina finite number of moves whenever II wins. That condition, topologically, is that the set A is closed. This fact—thatall closed games are determined—is called the Gale-Stewart theorem. Note that by symmetry, all open games aredetermined as well. (A game is open if I can win only by winning in a finite number of moves.)

7.3 Determinacy from ZFC

Gale and Stewart proved the open and closed games are determined. Determinacy for second level of the Borelhierarchy games was shown by Wolfe in 1955. Over the following 20 years, additional research using ever-more-complicated arguments established that third and fourth levels of the Borel hierarchy are determined.In 1975, Donald A. Martin proved that all Borel games are determined; that is, if A is a Borel subset of Baire space,then GA is determined. This result, known as Borel determinacy, is the best possible determinacy result provable inZFC, in the sense that the determinacy of the next higher Wadge class is not provable in ZFC.In 1971, before Martin obtained his proof, Harvey Friedman showed that any proof of Borel determinacy must usethe axiom of replacement in an essential way, in order to iterate the powerset axiom transfinitely often. Friedman’swork gives a level-by-level result detailing how many iterations of the powerset axiom are necessary to guaranteedeterminacy at each level of the Borel hierarchy.

7.4. DETERMINACY AND LARGE CARDINALS 15

For every integer n, ZFC\P proves determinacy in the nth level of the difference hierarchy of 03 sets, but ZFC\P

does not prove that for every integer n nth level of the difference hierarchy of Π03 sets is determined. See reverse

mathematics for other relations between determinacy and subsystems of second-order arithmetic.

7.4 Determinacy and large cardinals

There is an intimate relationship between determinacy and large cardinals. In general, stronger large cardinal axiomsprove the determinacy of larger pointclasses, higher in theWadge hierarchy, and the determinacy of such pointclasses,in turn, proves the existence of inner models of slightly weaker large cardinal axioms than those used to prove thedeterminacy of the pointclass in the first place.

7.4.1 Measurable cardinals

It follows from the existence of a measurable cardinal that every analytic game (also called aΣ11 game) is determined,

or equivalently that every coanalytic (or Π11 ) game is determined. (See Projective hierarchy for definitions.)

Actually a measurable cardinal is more than enough. A weaker principle — the existence of 0# is sufficient to provecoanalytic determinacy, and a little bit more: The precise result is that the existence of 0# is equivalent to the deter-minacy of all levels of the difference hierarchy below the ω2 level, i.e. ω·n-Π1

1 determinacy for every n .From a measurable cardinal we can improve this very slightly to ω2-Π1

1 determinacy. From the existence of moremeasurable cardinals, one can prove the determinacy of more levels of the difference hierarchy over Π1

1.

Proof of Determinacy from Sharps

For every real number r, Σ11(r) determinacy is equivalent to existence of r#. To illustrate how large cardinals lead to

determinacy, here is a proof of Σ11(r) determinacy given existence of r#.

Let A be a Σ11(r) subset of the Baire space. A = p[T] for some tree T (constructible from r) on (ω, ω). (That is x∈A

iff from some y, ((x0, y0), (x1, y1), ...) is a path through T.)Given a partial play s, let Ts be the subtree of T consistent with s subject to max(y0,y1,...,y ₑ ₍ ₎−₁)<len(s). Theadditional condition ensures that Ts is finite.To prove that A is determined, define auxiliary game as follows:In addition to ordinary moves, player 2 must play a mapping of Ts into ordinals (below a sufficiently large ordinal κ)such that

• each new move extends the previous mapping and

• the ordering of the ordinals agrees with the Kleene-Brouwer order on Ts .

Recall that Kleene-Brouwer order is like lexicographical order except that if s properly extends t then s<t. It is awell-ordering iff the tree is well-founded.The auxiliary game is open. Proof: If player 2 does not lose at a finite stage, then the union of all Ts (which is thetree that corresponds to the play) is well-founded, and so the result of the non-auxiliary play is not in A.Thus, the auxiliary game is determined. Proof: By transfinite induction, for each ordinal α compute the set of positionswhere player 1 can force a win in α steps, where a position with player 2 to move is losing (for player 2) in α steps ifffor every move the resulting position is losing in less than α steps. One strategy for player 1 is to reduce α with eachposition (say picking the least α and breaking ties by picking the least move), and one strategy for player 2 is to pickthe least (actually any would work) move that does not lead to a position with an α assigned. Note that L(r) containsthe set of winning positions as well as the winning strategies given above.A winning strategy for player 2 in the original game leads to winning strategy in the auxiliary game: The subtree ofT corresponding to the winning strategy is well-founded, so player 2 can pick ordinals based on the Kleene-Brouwerorder of the tree. Also, trivially, a winning strategy for player 2 in the auxiliary game gives a winning strategy forplayer 2 in original game.

16 CHAPTER 7. DETERMINACY

It remains to show that using r#, the above-mentioned winning strategy for player 1 in the auxiliary game can beconverted into a winning strategy in the original game. If the auxiliary response uses only ordinals with indiscernibles,then (by indiscernibility) the moves by player 1 do not depend on the auxiliary moves, and so the strategy can beconverted into a strategy for the original game (since player 2 can hold out with indiscernibles for any finite numberof steps). Suppose that player 1 loses in the original game. Then, the tree corresponding to a play is well-founded.Therefore, player 2 can win the auxiliary game by using auxiliary moves based on the indiscernibles (since the ordertype of indiscernibles exceeds the Kleene-Brouwer order of the tree), which contradicts player 1 winning the auxiliarygame.

7.4.2 Woodin cardinals

If there is a Woodin cardinal with a measurable cardinal above it, then Π12 determinacy holds. More generally, if

there are nWoodin cardinals with a measurable cardinal above them all, then Π1 ₊₁ determinacy holds. From Π1 ₊₁determinacy, it follows that there is a transitive inner model containing nWoodin cardinals.∆1

2 (lightface) determinacy is equiconsistent with aWoodin cardinal. If∆12 determinacy holds, then for a Turing cone

of x (that is for every real x of sufficiently high Turing degree), L[x] satisfies OD-determinacy (that is determinacy ofgames on integers of length ω and ordinal-definable payoff), and in HODL[x] ωL[x]

2 is a Woodin cardinal.

7.4.3 Projective determinacy

If there are infinitely many Woodin cardinals, then projective determinacy holds; that is, every game whose winningcondition is a projective set is determined. From projective determinacy it follows that, for every natural number n,there is a transitive inner model which satisfies that there are nWoodin cardinals.

7.4.4 Axiom of determinacy

The axiom of determinacy, or AD, asserts that every two-player game of perfect information of length ω, in whichthe players play naturals, is determined.AD is provably false from ZFC; using the axiom of choice one may prove the existence of a non-determined game.However, if there are infinitely many Woodin cardinals with a measurable above them all, then L(R) is a model ofZF that satisfies AD.

7.5 Consequences of determinacy

7.5.1 Regularity properties for sets of reals

If A is a subset of Baire space such that the Banach-Mazur game for A is determined, then either II has a win-ning strategy, in which case A is meager, or I has a winning strategy, in which case A is comeager on some openneighborhood.This does not quite imply that A has the property of Baire, but it comes close: A simple modification of the argumentshows that if Γ is an adequate pointclass such that every game in Γ is determined, then every set of reals in Γ has theproperty of Baire.In fact this result is not optimal; by considering the unfolded Banach-Mazur game we can show that determinacy ofΓ (for Γ with sufficient closure properties) implies that every set of reals that is the projection of a set in Γ has theproperty of Baire. So for example the existence of a measurable cardinal implies Π1

1 determinacy, which in turnimplies that every Σ1

2 set of reals has the property of Baire.By considering other games, we can show thatΠ1n determinacy implies that every Σ1n₊₁ set of reals has the propertyof Baire, is Lebesgue measurable (in fact universally measurable) and has the perfect set property.

7.6. MORE GENERAL GAMES 17

7.5.2 Periodicity theorems

• The first periodicity theorem implies that, for every natural number n, if Δ1₂n₊₁ determinacy holds, thenΠ1₂n₊₁ and Σ1₂n₊₂ have the prewellordering property (and that Σ1₂n₊₁ andΠ1₂n₊₂ do not have the prewellorder-ing property, but rather have the separation property).

• The second periodicity theorem implies that, for every natural number n, if Δ1₂n₊₁ determinacy holds, thenΠ1₂n₊₁ and Σ1₂n have the scale property.[1] In particular, if projective determinacy holds, then every projectiverelation has a projective uniformization.

• The third periodicity theorem gives a sufficient condition for a game to have a definable winning strategy.

7.5.3 Applications to decidability of certain second-order theories

In 1969, Michael O. Rabin proved that the second-order theory of n successors is decidable. A key component ofthe proof requires showing determinacy of parity games, which lie in the third level of the Borel hierarchy.

7.5.4 Wadge determinacy

Wadge determinacy is the statement that for all pairs A,B of subsets of Baire space, the Wadge game G(A,B) isdetermined. Similarly for a pointclass Γ, Γ Wadge determinacy is the statement that for all sets A,B in Γ, the Wadgegame G(A,B) is determined.Wadge determinacy implies the semilinear ordering principle for the Wadge order. Another consequence of Wadgedeterminacy is the perfect set property.In general, Γ Wadge determinacy is a consequence of the determinacy of Boolean combinations of sets in Γ. In theprojective hierarchy,Π1

1 Wadge determinacy is equivalent toΠ11 determinacy, as proved by Harrington. This result

was extendend by Hjorth to prove thatΠ12 Wadge determinacy (and in fact the semilinear ordering principle forΠ1

2)already implies Π1

2 determinacy.

This subsection is still incomplete

7.6 More general gamesThis section is still to be written

7.6.1 Games in which the objects played are not natural numbers

This subsection is incomplete.

Determinacy of games on ordinals with ordinal definable payoff and length ω implies that for every regular cardinalκ>ω there are no ordinal definable disjoint stationary subsets of κ made of ordinals of cofinality ω. The consistencystrength of the determinacy hypothesis is unknown but is expected to be very high.

7.6.2 Games played on trees

This subsection is still to be written

7.6.3 Long games

Existence of ω1 Woodin cardinals implies that for every countable ordinal α, all games on integers of length α andprojective payoff are determined. Roughly speaking, α Woodin cardinals corresponds to determinacy of games onreals of length α (with a simple payoff set). Assuming a limit of Woodin cardinals κ with o(κ)=κ++ and ω Woodincardinals above κ, games of variable countable length where the game ends as soon as its length is admissible relative

18 CHAPTER 7. DETERMINACY

to the line of play and with projective payoff are determined. Assuming that a certain iterability conjecture is provable,existence of a measurable Woodin cardinal implies determinacy of open games of length ω1 and projective payoff.(In these games, a winning condition for the first player is triggered at a countable stage, so the payoff can be codedas a set of reals.)Relative to a Woodin limit of Woodin cardinals and a measurable above them, it is consistent that every game onintegers of length ω1 and ordinal definable payoff is determined. It is conjectured that the determinacy hypothesisis equiconsistent with a Woodin limit of Woodin cardinals. ω1 is maximal in that there are undetermined games onintegers of length ω1+ω and ordinal definable payoff.

7.6.4 Games of imperfect information

In any interesting game with imperfect information, a winning strategy will be a mixed strategy: that is, it willgive some probability of differing responses to the same situation. If both players’ optimal strategies are mixedstrategies then the outcome of the game cannot be certainly determinant (as it can for pure strategies, since these aredeterministic). But the probability distribution of outcomes to opposing mixed strategies can be calculated. A gamethat requires mixed strategies is defined as determined if a strategy exists that yields a minimum expected value (overpossible counter-strategies) that exceeds a given value. Against this definition, all finite two player zero-sum gamesare clearly determined. However, the determinacy of infinite games of imperfect information (Blackwell games) isless clear.[2]

In 1969 David Blackwell proved that some “infinite games with imperfect information” (now called “Blackwellgames”) are determined, and in 1998 Donald A.Martin proved that ordinary (perfect-information game) determinacyfor a boldface pointclass implies Blackwell determinacy for the pointclass. This, combined with the Borel determi-nacy theorem of Martin, implies that all Blackwell games with Borel payoff functions are determined.[3] [4] Martinconjectured that ordinary determinacy and Blackwell determinacy for infinite games are equivalent in a strong sense(i.e. that Blackwell determinacy for a boldface pointclass in turn implies ordinary determinacy for that pointclass),but as of 2010, it has not been proven that Blackwell determinacy implies perfect-information-game determinacy.[5]

7.7 Quasistrategies and quasideterminacy

7.8 Footnotes[1] “Determinacy Maximum”. mit.edu.

[2] Vervoort, M. R. (1996). “Blackwell Games” (PDF). Statistics, Probability andGameTheory 30: 4& 5. doi:10.1214/lnms/1215453583.

[3] Martin, D. A. (December 1998). “The determinacy of Blackwell games”. Journal of Symbolic Logic 63 (4): 1565.doi:10.2307/2586667.

[4] Shmaya, E. (2009). “The determinacy of infinite games with eventual perfect monitoring” 30. arXiv:0902.2254.

[5] Benedikt Löwe (2006). “SET THEORY OF INFINITE IMPERFECT INFORMATION”. CiteSeerX. Retrieved 2010-06-06.

1. ^ This assumes that I is trying to get the intersection of neighborhoods played to be a singleton whose uniqueelement is an element of A. Some authors make that the goal instead for player II; that usage requires modifyingthe above remarks accordingly.

7.9 References

• Gale, D. and F. M. Stewart (1953). Kuhn, H. W.; Tucker, A. W., eds. “Contributions to the Theory ofGames, Volume II”. Annals of Mathematics Studies 28. Princeton University Press. pp. 245–266. ISBN9780691079356. |chapter= ignored (help)

• Harrington, Leo (Jan 1978). “Analytic determinacy and 0#". The Journal of Symbolic Logic 43 (4): 685–693.doi:10.2307/2273508. JSTOR 2273508.

7.9. REFERENCES 19

• Hjorth, Greg (Jan 1996). "Π12Wadge degrees”. Annals of Pure andApplied Logic 77: 53–74. doi:10.1016/0168-

0072(95)00011-9.

• Jech, Thomas (2002). Set theory, third millennium edition (revised and expanded). Springer. ISBN 3-540-44085-2.

• Martin, Donald A. (1975). “Borel determinacy”. Annals of Mathematics. Second Series 102 (2): 363–371.doi:10.2307/1971035.

• Martin, Donald A. and John R. Steel (Jan 1989). “A Proof of Projective Determinacy”. Journal of the Amer-ican Mathematical Society 2 (1): 71–125. doi:10.2307/1990913. JSTOR 1990913.

• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.

• Woodin, W. Hugh (1988). “Supercompact cardinals, sets of reals, and weakly homogeneous trees”. Proceed-ings of the National Academy of Sciences of theUnited States of America 85 (18): 6587–6591. doi:10.1073/pnas.85.18.6587.PMC 282022. PMID 16593979.

• Martin, Donald A. (2003). “A simple proof that determinacy implies Lebesgue measurability”. Rend. Sem.Mat. Univ. Pol. Torino 61 (4): 393–399. (PDF)

• Wolfe, P. (1955). “The strict determinateness of certain infinite games”. Pacific J. Math. 5: SupplementI:841–847. doi:10.2140/pjm.1955.5.841.

Chapter 8

Homogeneous tree

In descriptive set theory, a tree over a product set Y ×Z is said to be homogeneous if there is a system of measures⟨µs | s ∈ <ωY ⟩ such that the following conditions hold:

• µs is a countably-additive measure on {t | ⟨s, t⟩ ∈ T} .

• The measures are in some sense compatible under restriction of sequences: if s1 ⊆ s2 , then µs1(X) =1 ⇐⇒ µs2({t | t ↾ lh(s1) ∈ X}) = 1 .

• If x is in the projection of T , the ultrapower by ⟨µx↾n | n ∈ ω⟩ is wellfounded.

An equivalent definition is produced when the final condition is replaced with the following:

• There are ⟨µs | s ∈ ωY ⟩ such that if x is in the projection of [T ] and ∀n ∈ ω µx↾n(Xn) = 1 , then there isf ∈ ωZ such that ∀n ∈ ω f ↾ n ∈ Xn . This condition can be thought of as a sort of countable completenesscondition on the system of measures.

T is said to be κ -homogeneous if each µs is κ -complete.Homogeneous trees are involved in Martin and Steel's proof of projective determinacy.

8.1 References• Martin, Donald A. and John R. Steel (Jan 1989). “A Proof of Projective Determinacy”. Journal of the Amer-ican Mathematical Society (Journal of the American Mathematical Society, Vol. 2, No. 1) 2 (1): 71–125.doi:10.2307/1990913. JSTOR 1990913.

20

Chapter 9

Homogeneously Suslin set

In descriptive set theory, a set S is said to be homogeneously Suslin if it is the projection of a homogeneous tree. Sis said to be κ -homogeneously Suslin if it is the projection of a κ -homogeneous tree.If A ⊆ ωω is a 1

1 set and κ is a measurable cardinal, then A is κ -homogeneously Suslin. This result is important inthe proof that the existence of a measurable cardinal implies that 1

1 sets are determined.

9.1 See also• Projective determinacy

9.2 References• Martin, Donald A. and John R. Steel (Jan 1989). “A Proof of Projective Determinacy”. Journal of the Amer-ican Mathematical Society (American Mathematical Society) 2 (1): 71–125. doi:10.2307/1990913. JSTOR1990913.

21

Chapter 10

L(R)

In set theory, L(R) (pronounced L of R) is the smallest transitive inner model of ZF containing all the ordinals andall the reals.

10.1 Construction

It can be constructed in a manner analogous to the construction of L (that is, Gödel’s constructible universe), byadding in all the reals at the start, and then iterating the definable powerset operation through all the ordinals.

10.2 Assumptions

In general, the study of L(R) assumes a wide array of large cardinal axioms, since without these axioms one cannotshow even that L(R) is distinct from L. But given that sufficient large cardinals exist, L(R) does not satisfy the axiomof choice, but rather the axiom of determinacy. However, L(R) will still satisfy the axiom of dependent choice, givenonly that the von Neumann universe, V, also satisfies that axiom.

10.3 Results

Some additional results of the theory are:

• Every projective set of reals -- and therefore every analytic set and every Borel set of reals -- is an element ofL(R).

• Every set of reals in L(R) is Lebesgue measurable (in fact, universally measurable) and has the property ofBaire and the perfect set property.

• L(R) does not satisfy the axiom of uniformization or the axiom of real determinacy.

• R#, the sharp of the set of all reals, has the smallest Wadge degree of any set of reals not contained in L(R).

• While not every relation on the reals in L(R) has a uniformization in L(R), every such relation does have auniformization in L(R#).

• Given any (set-size) generic extension V[G] of V, L(R) is an elementary submodel of L(R) as calculated inV[G]. Thus the theory of L(R) cannot be changed by forcing.

• L(R) satisfies AD+.

22

10.4. REFERENCES 23

10.4 References• Woodin, W. Hugh (1988). “Supercompact cardinals, sets of reals, and weakly homogeneous trees”. Proceed-ings of the National Academy of Sciences of theUnited States of America 85 (18): 6587–6591. doi:10.1073/pnas.85.18.6587.PMC 282022. PMID 16593979.

Chapter 11

Lightface analytic game

In descriptive set theory, a lightface analytic game is a game whose payoff set A is a Σ11 subset of Baire space; that

is, there is a tree T on ω × ω which is a computable subset of (ω × ω)<ω , such that A is the projection of the set ofall branches of T.The determinacy of all lightface analytic games is equivalent to the existence of 0#.

24

Chapter 12

Martin measure

In descriptive set theory, the Martin measure is a filter on the set of Turing degrees of sets of natural numbers,named after Donald A. Martin. Under the axiom of determinacy it can be shown to be an ultrafilter.

12.1 Definition

Let D be the set of Turing degrees of sets of natural numbers. Given some equivalence class [X] ∈ D , we maydefine the cone (or upward cone) of [X] as the set of all Turing degrees [Y ] such that X ≤T Y ; that is, the set ofTuring degrees which are “more complex” than X under Turing reduction.We say that a set A of Turing degrees has measure 1 under the Martin measure exactly when A contains some cone.Since it is possible, for any A , to construct a game in which player I has a winning strategy exactly when A containsa cone and in which player II has a winning strategy exactly when the complement of A contains a cone, the axiomof determinacy implies that the measure-1 sets of Turing degrees form an ultrafilter.

12.2 Consequences

It is easy to show that a countable intersection of cones is itself a cone; the Martin measure is therefore a countablycomplete filter. This fact, combined with the fact that the Martin measure may be transferred to ω1 by a simplemapping, tells us that ω1 is measurable under the axiom of determinacy. This result shows part of the importantconnection between determinacy and large cardinals.

12.3 References• Moschovakis, Yiannis N. (2009). Descriptive Set Theory. Mathematical surveys and monographs 155 (2nded.). American Mathematical Society. p. 338. ISBN 9780821848135.

25

Chapter 13

Measurable cardinal

In mathematics, ameasurable cardinal is a certain kind of large cardinal number. In order to define the concept, oneintroduces a two-valued measure on a cardinal κ, or more generally on any set. For a cardinal κ, it can be describedas a subdivision of all of its subsets into large and small sets such that κ itself is large, ∅ and all singletons {α}, α ∈κ are small, complements of small sets are large and vice versa. The intersection of fewer than κ large sets is againlarge.[1]

It turns out that uncountable cardinals endowed with a two-valued measure are large cardinals whose existence cannotbe proved from ZFC.[2]

The concept of a measurable cardinal was introduced by Stanislaw Ulam in 1930.[3]

13.1 Definition

Formally, a measurable cardinal is an uncountable cardinal number κ such that there exists a κ-additive, non-trivial,0-1-valued measure on the power set of κ. (Here the term κ-additive means that, for any sequence Aα, α<λ ofcardinality λ<κ, Aα being pairwise disjoint sets of ordinals less than κ, the measure of the union of the Aα equals thesum of the measures of the individual Aα.)Equivalently, κ is measurable means that it is the critical point of a non-trivial elementary embedding of the universeV into a transitive class M. This equivalence is due to Jerome Keisler and Dana Scott, and uses the ultrapowerconstruction from model theory. Since V is a proper class, a technical problem that is not usually present whenconsidering ultrapowers needs to be addressed, by what is now called Scott’s trick.Equivalently, κ is a measurable cardinal if and only if it is an uncountable cardinal with a κ-complete, non-principalultrafilter. Again, this means that the intersection of any strictly less than κ-many sets in the ultrafilter, is also in theultrafilter.

13.2 Properties

Although it follows from ZFC that every measurable cardinal is inaccessible (and is ineffable, Ramsey, etc.), it isconsistent with ZF that a measurable cardinal can be a successor cardinal. It follows from ZF + axiom of determinacythat ω1 is measurable, and that every subset of ω1 contains or is disjoint from a closed and unbounded subset.Ulam showed that the smallest cardinal κ that admits a non-trivial countably-additive two-valued measure must infact admit a κ-additive measure. (If there were some collection of fewer than κ measure-0 subsets whose union wasκ, then the induced measure on this collection would be a counterexample to the minimality of κ.) From there, onecan prove (with the Axiom of Choice) that the least such cardinal must be inaccessible.It is trivial to note that if κ admits a non-trivial κ-additive measure, then κ must be regular. (By non-triviality andκ-additivity, any subset of cardinality less than κ must have measure 0, and then by κ-additivity again, this meansthat the entire set must not be a union of fewer than κ sets of cardinality less than κ.) Finally, if λ < κ, then it can't bethe case that κ ≤ 2λ. If this were the case, then we could identify κ with some collection of 0-1 sequences of lengthλ. For each position in the sequence, either the subset of sequences with 1 in that position or the subset with 0 in that

26

13.3. REAL-VALUED MEASURABLE 27

position would have to have measure 1. The intersection of these λ-many measure 1 subsets would thus also have tohave measure 1, but it would contain exactly one sequence, which would contradict the non-triviality of the measure.Thus, assuming the Axiom of Choice, we can infer that κ is a strong limit cardinal, which completes the proof of itsinaccessibility.If κ is measurable and p∈Vκ and M (the ultrapower of V) satisfies ψ(κ,p), then the set of α<κ such that V satisfiesψ(α,p) is stationary in κ (actually a set of measure 1). In particular if ψ is a Π1 formula and V satisfies ψ(κ,p), thenM satisfies it and thus V satisfies ψ(α,p) for a stationary set of α<κ. This property can be used to show that κ is alimit of most types of large cardinals which are weaker than measurable. Notice that the ultrafilter or measure whichwitnesses that κ is measurable cannot be inM since the smallest such measurable cardinal would have to have anothersuch below it which is impossible.Every measurable cardinal κ is a 0-huge cardinal because κM⊂M, that is, every function from κ to M is in M. Con-sequently, Vκ₊₁⊂M.

13.3 Real-valued measurable

A cardinal κ is called real-valued measurable if there is a κ-additive probability measure on the power set of κwhich vanishes on singletons. Real-valued measurable cardinals were introduced by Stefan Banach (1930). Banach& Kuratowski (1929) showed that the continuum hypothesis implies that c is not real-valued measurable. StanislawUlam (1930) showed that real valued measurable cardinals are weakly inaccessible (they are in fact weakly Mahlo).All measurable cardinals are real-valued measurable, and a real-valued measurable cardinal κ is measurable if andonly if κ is greater than c . Thus a cardinal is measurable if and only if it is real-valued measurable and stronglyinaccessible. A real valued measurable cardinal less than or equal to c exists if and only if there is a countablyadditive extension of the Lebesgue measure to all sets of real numbers if and only if there is an atomless probabilitymeasure on the power set of some non-empty set.Solovay (1971) showed that existence of measurable cardinals in ZFC, real valued measurable cardinals in ZFC, andmeasurable cardinals in ZF, are equiconsistent.

13.4 See also

• Normal measure

• Mitchell order

13.5 Notes[1] Maddy 1988

[2] Jech 2002

[3] Ulam 1930

13.6 References

• Banach, Stefan (1930), "Über additive Maßfunktionen in abstrakten Mengen”, Fundamenta Mathematicae 15:97–101, ISSN 0016-2736

• Banach, Stefan; Kuratowski, C. (1929), “Sur une généralisation du probleme de la mesure”, Fundamenta Math-ematicae 14: 127–131, ISSN 0016-2736

• Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations ofMathematics ; V. 76). Elsevier Science Ltd. ISBN 978-0-7204-2279-5.

28 CHAPTER 13. MEASURABLE CARDINAL

• Jech, Thomas (2002). Set theory, third millennium edition (revised and expanded). Springer. ISBN 3-540-44085-2.

• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nded.). Springer. ISBN 3-540-00384-3.

• Maddy, P. (1988). “Believeing the Axioms I”. J. Symb. Logic 53 (2): 481–511. doi:10.2307/2274569. (acopy (parts I & II) with corrections is available at the authors web page)

• Solovay, Robert M. (1971), “Real-valued measurable cardinals”, Axiomatic set theory (Proc. Sympos. PureMath., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967), Providence, R.I.: Amer. Math. Soc., pp.397–428, MR 0290961

• Ulam, Stanislaw (1930), “Zur Masstheorie in der allgemeinen Mengenlehre”, Fundamenta Mathematicae 16:140–150, ISSN 0016-2736

Chapter 14

Property of Baire

A subset A of a topological space X has the property of Baire (Baire property, named after René-Louis Baire),or is called an almost open set, if it differs from an open set by a meager set; that is, if there is an open set U ⊆ Xsuch that A∆U is meager (where Δ denotes the symmetric difference).[1]

The family of sets with the property of Baire forms a σ-algebra. That is, the complement of an almost open set isalmost open, and any countable union or intersection of almost open sets is again almost open.[1] Since every openset is almost open (the empty set is meager), it follows that every Borel set is almost open.If a subset of a Polish space has the property of Baire, then its corresponding Banach-Mazur game is determined.The converse does not hold; however, if every game in a given adequate pointclass Γ is determined, then every set inΓ has the property of Baire. Therefore it follows from projective determinacy, which in turn follows from sufficientlarge cardinals, that every projective set (in a Polish space) has the property of Baire.[2]

It follows from the axiom of choice that there are sets of reals without the property of Baire. In particular, the Vitaliset does not have the property of Baire.[3] Already weaker versions of choice are sufficient: the Boolean prime idealtheorem implies that there is a nonprincipal ultrafilter on the set of natural numbers; each such ultrafilter induces, viabinary representations of reals, a set of reals without the Baire property.[4]

14.1 See also• Baire category theorem

14.2 References[1] Oxtoby, John C. (1980), “4. The Property of Baire”, Measure and Category, Graduate Texts in Mathematics 2 (2nd ed.),

Springer-Verlag, pp. 19–21, ISBN 978-0-387-90508-2.

[2] Becker, Howard; Kechris, Alexander S. (1996), The descriptive set theory of Polish group actions, London MathematicalSociety Lecture Note Series 232, Cambridge University Press, Cambridge, p. 69, doi:10.1017/CBO9780511735264,ISBN 0-521-57605-9, MR 1425877.

[3] Oxtoby (1980), p. 22.

[4] Blass, Andreas (2010), “Ultrafilters and set theory”,Ultrafilters across mathematics, ContemporaryMathematics 530, Prov-idence, RI: American Mathematical Society, pp. 49–71, doi:10.1090/conm/530/10440, MR 2757533. See in particularp. 64.

14.3 External links• Springer Encyclopaedia of Mathematics article on Baire property

29

Chapter 15

Rank-into-rank

In set theory, a branch of mathematics, a rank-into-rank is a large cardinal λ satisfying one of the following fouraxioms given in order of increasing consistency strength. (They are sometimes known as rank-into-rank embeddings,where a rank is one of the sets Vλ of the von Neumann hierarchy.)

• Axiom I3: There is a nontrivial elementary embedding of Vλ into itself.

• Axiom I2: There is a nontrivial elementary embedding of V into a transitive class M that includes Vλ whereλ is the first fixed point above the critical point.

• Axiom I1: There is a nontrivial elementary embedding of Vλ₊₁ into itself.

• Axiom I0: There is a nontrivial elementary embedding of L(Vλ₊₁) into itself with the critical point below λ.

These are essentially the strongest known large cardinal axioms not known to be inconsistent in ZFC; the axiom forReinhardt cardinals is stronger, but is not consistent with the axiom of choice.If j is the elementary embedding mentioned in one of these axioms and κ is its critical point, then λ is the limit ofjn(κ) as n goes to ω. More generally, if the axiom of choice holds, it is provable that if there is a nontrivial elementaryembedding of Vα into itself then α is either a limit ordinal of cofinality ω or the successor of such an ordinal.The axioms I1, I2, and I3 were at first suspected to be inconsistent (in ZFC) as it was thought possible that Kunen’sinconsistency theorem that Reinhardt cardinals are inconsistent with the axiom of choice could be extended to them,but this has not yet happened and they are now usually believed to be consistent.Every I0 cardinal κ (speaking here of the critical point of j) is an I1 cardinal.Every I1 cardinal κ is an I2 cardinal and has a stationary set of I2 cardinals below it.Every I2 cardinal κ is an I3 cardinal and has a stationary set of I3 cardinals below it.Every I3 cardinal κ has another I3 cardinal above it and is an n-huge cardinal for every n<ω.Axiom I1 implies that Vλ₊₁ (equivalently, H(λ+)) does not satisfy V=HOD. There is no set S⊂λ definable in Vλ₊₁ (evenfrom parameters Vλ and ordinals <λ+) with S cofinal in λ and |S|<λ, that is, no such S witnesses that λ is singular.And similarly for Axiom I0 and ordinal definability in L(Vλ₊₁) (even from parameters in Vλ). However globally, andeven in Vλ,[1] V=HOD is relatively consistent with Axiom I1.

15.1 References

• Gaifman, Haim (1974), “Elementary embeddings of models of set-theory and certain subtheories”, Axiomaticset theory, Proc. Sympos. Pure Math., XIII, Part II, Providence R.I.: Amer. Math. Soc., pp. 33–101, MR0376347

• Kanamori, Akihiro (2003), The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nded.), Springer, ISBN 3-540-00384-3

30

15.1. REFERENCES 31

• Laver, Richard (1997), “Implications between strong large cardinal axioms”, Ann. Pure Appl. Logic 90 (1-3):79–90, doi:10.1016/S0168-0072(97)00031-6, MR 1489305

• Solovay, Robert M.; Reinhardt, William N.; Kanamori, Akihiro (1978), “Strong axioms of infinity and ele-mentary embeddings”, Annals of Mathematical Logic 13 (1): 73–116, doi:10.1016/0003-4843(78)90031-1

[1] Consistency of V = HOD With the Wholeness Axiom, Paul Corazza, Archive for Mathematical Logic, No. 39, 2000.

Chapter 16

Tree (descriptive set theory)

This article is about mathematical trees described by prefixes of finite sequences. For trees described by partiallyordered sets, see Tree (set theory).

In descriptive set theory, a tree on a setX is a collection of finite sequences of elements ofX such that every prefixof a sequence in the collection also belongs to the collection.

16.1 Definitions

16.1.1 Trees

The collection of all finite sequences of elements of a setX is denotedX<ω . With this notation, a tree is a nonemptysubset T of X<ω , such that if ⟨x0, x1, . . . , xn−1⟩ is a sequence of length n in T , and if 0 ≤ m < n , then theshortened sequence ⟨x0, x1, . . . , xm−1⟩ also belongs to T . In particular, choosing m = 0 shows that the emptysequence belongs to every tree.

16.1.2 Branches and bodies

A branch through a tree T is an infinite sequence of elements ofX , each of whose finite prefixes belongs to T . Theset of all branches through T is denoted [T ] and called the body of the tree T .A tree that has no branches is called wellfounded; a tree with at least one branch is illfounded. By König’s lemma, atree on a finite set with an infinite number of sequences must necessarily be illfounded.

16.1.3 Terminal nodes

A finite sequence that belongs to a tree T is called a terminal node if it is not a prefix of a longer sequence in T .Equivalently, ⟨x0, x1, . . . , xn−1⟩ ∈ T is terminal if there is no element x ofX such that that ⟨x0, x1, . . . , xn−1, x⟩ ∈T . A tree that does not have any terminal nodes is called pruned.

16.2 Relation to other types of trees

In graph theory, a rooted tree is a directed graph in which every vertex except for a special root vertex has exactlyone outgoing edge, and in which the path formed by following these edges from any vertex eventually leads to theroot vertex. If T is a tree in the descriptive set theory sense, then it corresponds to a graph with one vertex for eachsequence in T , and an outgoing edge from each nonempty sequence that connects it to the shorter sequence formedby removing its last element. This graph is a tree in the graph-theoretic sense. The root of the tree is the emptysequence.

32

16.3. TOPOLOGY 33

In order theory, a different notion of a tree is used: an order-theoretic tree is a partially ordered set with one minimalelement in which each element has a well-ordered set of predecessors. Every tree in descriptive set theory is also anorder-theoretic tree, using a partial ordering in which two sequences T and U are ordered by T < U if and only ifT is a proper prefix of U . The empty sequence is the unique minimal element, and each element has a finite andwell-ordered set of predecessors (the set of all of its prefixes). An order-theoretic tree may be represented by anisomorphic tree of sequences if and only if each of its elements has finite height (that is, a finite set of predecessors).

16.3 Topology

The set of infinite sequences over X (denoted as Xω ) may be given the product topology, treating X as a discretespace. In this topology, every closed subset C of Xω is of the form [T ] for some pruned tree T . Namely, let Tconsist of the set of finite prefixes of the infinite sequences in C . Conversely, the body [T ] of every tree T forms aclosed set in this topology.Frequently trees on Cartesian productsX ×Y are considered. In this case, by convention, the set of finite sequencesof members of the product space, (X × Y )<ω , is identified in the natural way with a subset of the product of twospaces of sequences,X<ω × Y <ω (the subset of members of the second product for which both sequences have thesame length). In this way a tree [T ] over the product space may be considered as a subset ofX<ω × Y <ω . We maythen form the projection of [T ] ,

p[T ] = {x⃗ ∈ Xω|(∃y⃗ ∈ Y ω)⟨x⃗, y⃗⟩ ∈ [T ]}

16.4 See also• Laver tree, a type of tree used in set theory as part of a notion of forcing

16.5 References• Kechris, Alexander S. (1995). Classical Descriptive Set Theory. Graduate Texts in Mathematics 156. Springer.ISBN 0-387-94374-9 ISBN 3-540-94374-9.

Chapter 17

Universally measurable set

In mathematics, a subset A of a Polish space X is universally measurable if it is measurable with respect to everycomplete probability measure onX that measures all Borel subsets ofX . In particular, a universally measurable setof reals is necessarily Lebesgue measurable (see #Finiteness condition) below.Every analytic set is universally measurable. It follows from projective determinacy, which in turn follows fromsufficient large cardinals, that every projective set is universally measurable.

17.1 Finiteness condition

The condition that the measure be a probability measure; that is, that the measure of X itself be 1, is less restrictivethan it may appear. For example, Lebesgue measure on the reals is not a probability measure, yet every universallymeasurable set is Lebesgue measurable. To see this, divide the real line into countably many intervals of length 1; say,N0=[0,1), N1=[1,2), N2=[−1,0), N3=[2,3), N4=[−2,−1), and so on. Now letting μ be Lebesgue measure, define anew measure ν by

ν(A) =∞∑i=0

1

2n+1µ(A ∩Ni)

Then easily ν is a probability measure on the reals, and a set is ν-measurable if and only if it is Lebesgue measurable.More generally a universally measurable set must be measurable with respect to every sigma-finite measure thatmeasures all Borel sets.

17.2 Example contrasting with Lebesgue measurability

Suppose A is a subset of Cantor space 2ω ; that is, A is a set of infinite sequences of zeroes and ones. By putting abinary point before such a sequence, the sequence can be viewed as a real number between 0 and 1 (inclusive), withsome unimportant ambiguity. Thus we can think of A as a subset of the interval [0,1], and evaluate its Lebesguemeasure. That value is sometimes called the coin-flipping measure of A , because it is the probability of producinga sequence of heads and tails that is an element of A , upon flipping a fair coin infinitely many times.Now it follows from the axiom of choice that there are some such A without a well-defined Lebesgue measure (orcoin-flipping measure). That is, for such an A , the probability that the sequence of flips of a fair coin will wind upin A is not well-defined. This is a pathological property of A that says that A is “very complicated” or “ill-behaved”.From such a set A , form a new set A′ by performing the following operation on each sequence in A : Interspersea 0 at every even position in the sequence, moving the other bits to make room. Now A′ is intuitively no “simpler”or “better-behaved” than A . However, the probability that the sequence of flips of a fair coin will wind up in A′ iswell-defined, for the rather silly reason that the probability is zero (to get into A′ , the coin must come up tails onevery even-numbered flip).

34

17.3. REFERENCES 35

For such a set of sequences to be universallymeasurable, on the other hand, an arbitrarily biased coin may be used—even one that can “remember” the sequence of flips that has gone before—and the probability that the sequence ofits flips ends up in the set, must be well-defined. Thus the A′ described above is not universally measurable, becausewe can test it against a coin that always comes up tails on even-numbered flips, and is fair on odd-numbered flips.

17.3 References• Alexander Kechris (1995), Classical Descriptive Set Theory, Graduate Texts in Mathematics 156, Springer,ISBN 0-387-94374-9

• Nishiura Togo (2008), Absolute Measurable Spaces, Cambridge University Press, ISBN 0-521-87556-0

Chapter 18

Woodin cardinal

In set theory, aWoodin cardinal (named for W. Hugh Woodin) is a cardinal number λ such that for all functions

f : λ → λ

there exists a cardinal κ < λ with

{f(β)|β < κ} ⊆ κ

and an elementary embedding

j : V → M

from the Von Neumann universe V into a transitive inner model M with critical point κ and

V ₍ ₎₍κ₎ ⊆ M.

An equivalent definition is this: λ is Woodin if and only if λ is strongly inaccessible and for all A ⊆ Vλ there exists aλA < λ which is < λ - A -strong.λA being < λ - A -strong means that for all ordinals α < λ, there exist a j : V → M which is an elementaryembedding with critical point λA , j(λA) > α , Vα ⊆ M and j(A) ∩ Vα = A ∩ Vα . (See also strong cardinal.)A Woodin cardinal is preceded by a stationary set of measurable cardinals, and thus it is a Mahlo cardinal. However,the first Woodin cardinal is not even weakly compact.

18.1 Consequences

Woodin cardinals are important in descriptive set theory. By a result[1] of Martin and Steel, existence of infinitelymany Woodin cardinals implies projective determinacy, which in turn implies that every projective set is measurable,has the Baire property (differs from an open set by a meager set, that is, a set which is a countable union of nowheredense sets), and the perfect set property (is either countable or contains a perfect subset).The consistency of the existence of Woodin cardinals can be proved using determinacy hypotheses. Working inZF+AD+DC one can prove that Θ0 is Woodin in the class of hereditarily ordinal-definable sets. Θ0 is the firstordinal onto which the continuum cannot be mapped by an ordinal-definable surjection (see Θ (set theory)).Shelah proved that if the existence of a Woodin cardinal is consistent then it is consistent that the nonstationary idealon ω1 is ℵ2 -saturated. Woodin also proved the equiconsistency of the existence of infinitely many Woodin cardinalsand the existence of an ℵ1 -dense ideal over ℵ1 .

36

18.2. HYPER-WOODIN CARDINALS 37

18.2 Hyper-Woodin cardinals

A cardinal κ is called hyper-Woodin if there exists a normal measure U on κ such that for every set S, the set

{λ < κ | λ is <κ-S-strong}

is in U.λ is <κ-S-strong if and only if for each δ < κ there is a transitive class N and an elementary embedding

j : V → N

with

λ = crit(j),j(λ)≥ δ, and

j(S) ∩Hδ = S ∩Hδ

The name alludes to the classical result that a cardinal is Woodin if and only if for every set S, the set

{λ < κ | λ is <κ-S-strong}

is a stationary setThe measure U will contain the set of all Shelah cardinals below κ.

18.3 Weakly hyper-Woodin cardinals

A cardinal κ is called weakly hyper-Woodin if for every set S there exists a normal measure U on κ such that the set{λ < κ | λ is <κ-S-strong} is in U. λ is <κ-S-strong if and only if for each δ < κ there is a transitive class N and anelementary embedding j : V → N with λ = crit(j), j(λ) >= δ, and j(S) ∩Hδ = S ∩Hδ.

The name alludes to the classic result that a cardinal is Woodin if for every set S, the set {λ < κ | λ is <κ-S-strong} isstationary.The difference between hyper-Woodin cardinals and weakly hyper-Woodin cardinals is that the choice of U does notdepend on the choice of the set S for hyper-Woodin cardinals.

18.4 Notes and references[1] A Proof of Projective Determinacy

18.5 Further reading• Kanamori, Akihiro (2003). The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (2nded.). Springer. ISBN 3-540-00384-3.

• For proofs of the two results listed in consequences see Handbook of Set Theory (Eds. Foreman, Kanamori,Magidor) (to appear). Drafts of some chapters are available.

• Ernest Schimmerling, Woodin cardinals, Shelah cardinals and the Mitchell-Steel core model, Proceedings ofthe American Mathematical Society 130/11, pp. 3385–3391, 2002, online

• Steel, John R. (October 2007). “What is a Woodin Cardinal?" (PDF). Notices of the American MathematicalSociety 54 (9): 1146–7. Retrieved 2008-01-15.

Chapter 19

Zero sharp

In the mathematical discipline of set theory, 0# (zero sharp, also 0#) is the set of true formulae about indiscerniblesand order-indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers (usingGödel numbering), or as a subset of the hereditarily finite sets, or as a real number. Its existence is unprovable in ZFC,the standard form of axiomatic set theory, but follows from a suitable large cardinal axiom. It was first introduced asa set of formulae in Silver’s 1966 thesis, later published as Silver (1971), where it was denoted by Σ, and rediscoveredby Solovay (1967, p.52), who considered it as a subset of the natural numbers and introduced the notation O# (witha capital letter O; this later changed to a number 0).Roughly speaking, if 0# exists then the universe V of sets is much larger than the universe L of constructible sets,while if it does not exist then the universe of all sets is closely approximated by the constructible sets.

19.1 Definition

Zero sharp was defined by Silver and Solovay as follows. Consider the language of set theory with extra constantsymbols c1, c2, ... for each positive integer. Then 0# is defined to be the set of Gödel numbers of the true sentencesabout the constructible universe, with ci interpreted as the uncountable cardinal ℵi. (Here ℵi means ℵi in the fulluniverse, not the constructible universe.)There is a subtlety about this definition: by Tarski’s undefinability theorem it is not in general possible to define thetruth of a formula of set theory in the language of set theory. To solve this, Silver and Solovay assumed the existenceof a suitable large cardinal, such as a Ramsey cardinal, and showed that with this extra assumption it is possible todefine the truth of statements about the constructible universe. More generally, the definition of 0# works providedthat there is an uncountable set of indiscernibles for some Lα, and the phrase “0# exists” is used as a shorthand wayof saying this.There are several minor variations of the definition of 0#, which make no significant difference to its properties. Thereare many different choices of Gödel numbering, and 0# depends on this choice. Instead of being considered as a subsetof the natural numbers, it is also possible to encode 0# as a subset of formulae of a language, or as a subset of thehereditarily finite sets, or as a real number.

19.2 Statements that imply the existence of 0#

The condition about the existence of a Ramsey cardinal implying that 0# exists can be weakened. The existenceof ω1-Erdős cardinals implies the existence of 0#. This is close to being best possible, because the existence of 0#implies that in the constructible universe there is an α-Erdős cardinal for all countable α, so such cardinals cannot beused to prove the existence of 0#.Chang’s conjecture implies the existence of 0#.

38

19.3. STATEMENTS EQUIVALENT TO EXISTENCE OF 0# 39

19.3 Statements equivalent to existence of 0#

Kunen showed that 0# exists if and only if there exists a non-trivial elementary embedding for the Gödel constructibleuniverse L into itself.Donald A.Martin and Leo Harrington have shown that the existence of 0# is equivalent to the determinacy of lightfaceanalytic games. In fact, the strategy for a universal lightface analytic game has the same Turing degree as 0#.It follows from Jensen’s covering theorem that the existence of 0# is equivalent to ωω being a regular cardinal in theconstructible universe L.Silver showed that the existence of an uncountable set of indiscernibles in the constructible universe is equivalent tothe existence of 0#.

19.4 Consequences of existence and non-existence

Its existence implies that every uncountable cardinal in the set-theoretic universe V is an indiscernible in L andsatisfies all large cardinal axioms that are realized in L (such as being totally ineffable). It follows that the existenceof 0# contradicts the axiom of constructibility: V = L.If 0# exists, then it is an example of a non-constructible Δ13 set of integers. This is in some sense the simplest possibility for a non-constructible set, since all Σ12 and Π12 sets of integers are constructible.On the other hand, if 0# does not exist, then the constructible universe L is the core model—that is, the canonicalinner model that approximates the large cardinal structure of the universe considered. In that case, Jensen’s coveringlemma holds:

For every uncountable set x of ordinals there is a constructible y such that x ⊂ y and y has the samecardinality as x.

This deep result is due to Ronald Jensen. Using forcing it is easy to see that the condition that x is uncountable cannotbe removed. For example, consider Namba forcing, that preserves ω1 and collapses ω2 to an ordinal of cofinality ω. Let G be an ω -sequence cofinal on ωL

2 and generic over L. Then no set in L of L-size smaller than ωL2 (which is

uncountable in V, since ω1 is preserved) can cover G , since ω2 is a regular cardinal.

19.5 Other sharps

If x is any set, then x# is defined analogously to 0# except that one uses L[x] instead of L. See the section on relativeconstructibility in constructible universe.

19.6 See also

• 0†, a set similar to 0# where the constructible universe is replaced by a larger inner model with a measurablecardinal.

19.7 References

• Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations ofMathematics ; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2.

• Harrington, Leo (1978), “Analytic determinacy and 0#", The Journal of Symbolic Logic 43 (4): 685–693,doi:10.2307/2273508, ISSN 0022-4812, MR 518675

40 CHAPTER 19. ZERO SHARP

• Jech, Thomas (2003). Set Theory. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, NewYork: Springer-Verlag. ISBN 978-3-540-44085-7. Zbl 1007.03002.

• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nded.). Springer. ISBN 3-540-00384-3.

• Martin, Donald A. (1970), “Measurable cardinals and analytic games”, Polska Akademia Nauk. FundamentaMathematicae 66: 287–291, ISSN 0016-2736, MR 0258637

• Silver, Jack H. (1971) [1966], “Some applications of model theory in set theory”, Annals of Pure and AppliedLogic 3 (1): 45–110, doi:10.1016/0003-4843(71)90010-6, ISSN 0168-0072, MR 0409188

• Solovay, Robert M. (1967), “A nonconstructible Δ13 set of integers”, Transactions of the American Mathematical Society 127: 50–75, ISSN 0002-9947, JSTOR1994631, MR 0211873

Chapter 20

Θ (set theory)

In set theory, Θ (pronounced like the letter theta) is the least nonzero ordinal α such that there is no surjection fromthe reals onto α.If the axiom of choice (AC) holds (or even if the reals can be wellordered) then Θ is simply (2ℵ0)+ , the cardinalsuccessor of the cardinality of the continuum. However, Θ is often studied in contexts where the axiom of choicefails, such as models of the axiom of determinacy.Θ is also the supremum of the lengths of all prewellorderings of the reals.

20.1 Proof of existence

It may not be obvious that it can be proven, without using AC, that there even exists a nonzero ordinal onto whichthere is no surjection from the reals (if there is such an ordinal, then there must be a least one because the ordinalsare wellordered). However, suppose there were no such ordinal. Then to every ordinal α we could associate the setof all prewellorderings of the reals having length α. This would give an injection from the class of all ordinals intothe set of all sets of orderings on the reals (which can to be seen to be a set via repeated application of the powersetaxiom). Now the axiom of replacement shows that the class of all ordinals is in fact a set. But that is impossible, bythe Burali-Forti paradox.

41

42 CHAPTER 20. Θ (SET THEORY)

20.2 Text and image sources, contributors, and licenses

20.2.1 Text• AD+ Source: https://en.wikipedia.org/wiki/AD%2B?oldid=646851881 Contributors: Charles Matthews, Jackol, MFH, Salix alba, Trova-

tore, SmackBot, RDBury, JoshuaZ, Mets501, Leyo, Hans Adler, Estudiarme, FrescoBot, Full-date unlinking bot and Anonymous: 1• Axiom of determinacy Source: https://en.wikipedia.org/wiki/Axiom_of_determinacy?oldid=669818717 Contributors: TakuyaMurata,

Schneelocke, Charles Matthews, Aleph4, Tobias Bergemann, Giftlite, Gene Ward Smith, Waltpohl, Leonard G., Barnaby dawson, MikeRosoft, Gauge, EmilJ, Oleg Alexandrov, Skoban, BD2412, Kbdank71, Salix alba, R.e.b., RussBot, Splash, Trovatore, Bandalism inBrogress, SmackBot, Atomota, Bluebot, Cthuljew, Ligulembot, Stotr~enwiki, Zero sharp, CRGreathouse, CmdrObot, CBM, Ntsimp,Kjs50, Dgianotti, Headbomb, David Eppstein, Kope, Leocat, Lwr314, Hesam7, Arcfrk, Phillist, Rtelgarsky, Addbot, DOI bot, Lightbot,Luckas-bot, AnomieBOT, Citation bot, Jsharpminor, Citation bot 1, Max Longint, Paolo Lipparini, BattyBot, Mark viking, Y2N1-09631and Anonymous: 13

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• Axiomof real determinacy Source: https://en.wikipedia.org/wiki/Axiom_of_real_determinacy?oldid=358402829Contributors: MichaelHardy, Schneelocke, Charles Matthews, Oleg Alexandrov, Trovatore, Bluebot, CBM, Hans Adler, Erik9bot and Anonymous: 1

• Banach–Mazur game Source: https://en.wikipedia.org/wiki/Banach%E2%80%93Mazur_game?oldid=618515338 Contributors: Axel-Boldt, Michael Hardy, Revolver, Charles Matthews, Chtito, Oleg Alexandrov, Kzollman, Graham87, R.e.b., CiaPan, Algebraist, GaiusCornelius, Trovatore, Kewp, SmackBot, Selfworm, Stotr~enwiki, Kjs50, Magioladitis, Althai, PbBot, Skeptical scientist, Rtelgarsky,Addbot, Yobot, Howard McCay, ZéroBot, ChuispastonBot and Anonymous: 8

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