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{{Short description|Possible axiom for set theory}}
In [[mathematics]], the '''axiom of determinacy''' (abbreviated as '''AD''') is a possible [[axiom]] for [[set theory]] introduced by [[Jan Mycielski]] and [[Hugo Steinhaus]] in 1962. It refers to certain two-person [[topological game]]s of length [[ω (ordinal number)|ω]]. AD states that every game of a [[Axiom of determinacy#Types of game that are determined|certain type]] is [[determined game|determined]]; that is, one of the two players has a [[winning strategy]].
In [[mathematics]], the '''axiom of determinacy''' (abbreviated as '''AD''') is a possible [[axiom]] for [[set theory]] introduced by [[Jan Mycielski]] and [[Hugo Steinhaus]] in 1962. It refers to certain two-person [[topological game]]s of length [[ω (ordinal number)|ω]]. AD states that every game of a [[Axiom of determinacy#Types of game that are determined|certain type]] is [[determined game|determined]]; that is, one of the two players has a [[winning strategy]].


They motivated AD by its interesting consequences, and suggested that AD could be true in the least natural model [[L(R)]] of a set theory, which accepts only a weak form of the [[axiom of choice]] (AC) but contains all [[real number|real]] and all [[ordinal number]]s. Some consequences of AD followed from theorems proved earlier by [[Stefan Banach]] and [[Stanisław Mazur]], and [[Morton Davis]]. [[Mycielski]] and [[Stanisław Świerczkowski]] contributed another one: AD implies that all sets of [[real number]]s are [[Lebesgue measurable]]. Later [[Donald A. Martin]] and others proved more important consequences, especially in [[descriptive set theory]]. In 1988, [[John R. Steel]] and [[W. Hugh Woodin]] concluded a long line of research. Assuming the existence of some [[uncountable]] [[cardinal number]]s analogous to <math>\alef_0</math>, they proved the original conjecture of Mycielski and Steinhaus that AD is true in L(R).
Steinhaus and Mycielski's motivation for AD was its interesting consequences, and suggested that AD could be true in the smallest natural model [[L(R)]] of a set theory, which accepts only a weak form of the [[axiom of choice]] (AC) but contains all [[real number|real]] and all [[ordinal number]]s. Some consequences of AD followed from theorems proved earlier by [[Stefan Banach]] and [[Stanisław Mazur]], and [[Morton Davis]]. [[Mycielski]] and [[Stanisław Świerczkowski]] contributed another one: AD implies that all sets of [[real number]]s are [[Lebesgue measurable]]. Later [[Donald A. Martin]] and others proved more important consequences, especially in [[descriptive set theory]]. In 1988, [[John R. Steel]] and [[W. Hugh Woodin]] concluded a long line of research. Assuming the existence of some [[Large cardinal|uncountable cardinal numbers analogous to <sub>0</sub>]], they proved the original conjecture of Mycielski and Steinhaus that AD is true in L(R).


==Types of game that are determined==
==Types of game that are determined==

The axiom of determinacy refers to games of the following specific form:
The axiom of determinacy refers to games of the following specific form:
Consider a subset ''A'' of the [[Baire space (set theory)|Baire space]] ''ω<sup>ω</sup>'' of all [[infinite sequence]]s of [[natural number]]s. Two players, '''I''' and '''II''', alternately pick natural numbers
Consider a subset ''A'' of the [[Baire space (set theory)|Baire space]] ω<sup>ω</sup> of all [[infinite sequence]]s of [[natural number]]s. Two players alternately pick natural numbers
:''n''<sub>0</sub>, ''n''<sub>1</sub>, ''n''<sub>2</sub>, ''n''<sub>3</sub>, ...
:''n''<sub>0</sub>, ''n''<sub>1</sub>, ''n''<sub>2</sub>, ''n''<sub>3</sub>,&nbsp;...
After infinitely many moves, a sequence <math>(n_i)_{i \in \omega}</math> is generated. Player '''I''' wins the game if and only if the sequence generated is an element of ''A''. The axiom of determinacy is the statement that all such games are determined.
That generates the sequence ⟨''n''<sub>''i''</sub>⟩<sub>''i''∈ω</sub> after infinitely many moves. The player who picks first wins the game if and only if the sequence generated is an element of ''A''. The axiom of determinacy is the statement that all such games are determined.


Not all games require the axiom of determinacy to prove them determined. If the set ''A'' is [[Clopen set|clopen]], the game is essentially a finite game, and is therefore determined. Similarly, if ''A'' is a [[closed set]], then the game is determined. It was shown in 1975 by [[Donald A. Martin]] that games whose winning set is a [[Borel set]] are determined. It follows from the existence of sufficiently [[large cardinal]]s that all games with winning set a [[projective set]] are determined (see [[Projective determinacy]]), and that AD holds in [[L(R)]].
Not all games require the axiom of determinacy to prove them determined. If the set ''A'' is [[clopen]], the game is essentially a finite game, and is therefore determined. Similarly, if ''A'' is a [[closed set]], then the game is determined. By the [[Borel determinacy theorem]], games whose winning set is a [[Borel set]] are determined. It follows from the existence of sufficiently [[large cardinal]]s that AD holds in [[L(R)]] and that a game is determined if it has a [[projective set]] as its winning set (see [[Projective determinacy]]).


The axiom of determinacy implies that for every subspace ''X'' of the [[Real line#As a topological space|real numbers]], the [[Banach–Mazur game]] ''BM''(''X'') is determined (and therefore that every set of reals has the [[property of Baire]]).
The axiom of determinacy implies that for every subspace ''X'' of the [[Real line#As a topological space|real numbers]], the [[Banach–Mazur game]] BM(''X'') is determined, and consequently, that every set of reals has the [[property of Baire]].


== Incompatibility of the axiom of determinacy with the axiom of choice ==
== Incompatibility with the axiom of choice ==
Under assumption of the axiom of choice, we present two separate constructions of counterexamples to the axiom of determinacy. It follows that the axiom of determinacy and the axiom of choice are incompatible.


=== Using a well-ordering of the continuum ===
The set S1 of all first player strategies in an ω-game ''G'' has the same [[cardinality]] as the [[cardinality of the continuum|continuum]]. The same is true of the set S2 of all second player strategies. We note that the cardinality of the set SG of all sequences possible in ''G'' is also the continuum. Let A be the subset of SG of all sequences that make the first player win. With the axiom of choice we can [[well order]] the continuum; furthermore, we can do so in such a way that any proper initial portion does not have the cardinality of the continuum. We create a counterexample by [[transfinite induction]] on the set of strategies under this well ordering:
The set ''S''<sub>1</sub> of all first player strategies in an ω-game ''G'' has the same [[cardinality]] as the [[cardinality of the continuum|continuum]]. The same is true for the set ''S''<sub>2</sub> of all second player strategies. Let ''SG'' be the set of all possible sequences in ''G'', and ''A'' be the subset of sequences of ''SG'' that make the first player win. With the axiom of choice we can [[well order]] the continuum, and we can do so in such a way that any proper initial portion has lower cardinality than the continuum. We use the obtained well ordered set ''J'' to index both ''S''<sub>1</sub> and ''S''<sub>2</sub>, and construct ''A'' such that it will be a counterexample.


We start with the set A undefined. Let T be the "time" whose axis has length continuum. We need to consider all strategies {s1(T)} of the first player and all strategies {s2(T)} of the second player to make sure that for every strategy there is a strategy of the other player that wins against it. For every strategy of the player considered we will generate a sequence that gives the other player a win. Let t be the time whose axis has length ℵ<sub>0</sub> and which is used during each game sequence.
We start with empty sets ''A'' and ''B''. Let ''α''&nbsp;∈&nbsp;''J'' be the index of the strategies in ''S''<sub>1</sub> and ''S''<sub>2</sub>. We need to consider all strategies ''S''<sub>1</sub>&nbsp;=&nbsp;{s<sub>1</sub>(''α'')}<sub>''α''∈''J''</sub> of the first player and all strategies ''S''<sub>2</sub>&nbsp;=&nbsp;{s<sub>2</sub>(''α'')}<sub>''α''∈''J''</sub> of the second player to make sure that for every strategy there is a strategy of the other player that wins against it. For every strategy of the player considered we will generate a sequence that gives the other player a win. Let ''t'' be the time whose axis has length ℵ<sub>0</sub> and which is used during each game sequence. We create the counterexample ''A'' by [[transfinite recursion]] on ''α'':


# Consider the current strategy {s1(T)} of the first player.
# Consider the strategy s<sub>1</sub>(''α'') of the first player.
# Apply this strategy on an ω-game, generating (together with the first player's strategy s<sub>1</sub>(''α'')) a sequence ⟨''a''<sub>1</sub>, ''b''<sub>2</sub>, ''a''<sub>3</sub>, ''b''<sub>4</sub>,&nbsp;...,a<sub>''t''</sub>, ''b''<sub>''t''+1</sub>,&nbsp;...⟩, which does not belong to ''A''. This is possible, because the number of choices for ⟨''b''<sub>2</sub>, ''b''<sub>4</sub>, ''b''<sub>6</sub>,&nbsp;...⟩ has the same cardinality as the continuum, which is larger than the cardinality of the proper initial portion {&nbsp;''β''&nbsp;∈&nbsp;''J''&nbsp;| ''β''&nbsp;<&nbsp;''α''&nbsp;} of ''J.''
#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),...}.
# Add this sequence to ''B'' to indicate that s<sub>1</sub>(''α'') loses (on ⟨''b''<sub>2</sub>, ''b''<sub>4</sub>, ''b''<sub>6</sub>,&nbsp;...⟩).
#Decide that this sequence does not belong to A, i.e. s1(T) lost.
#Consider the strategy {s2(T)} of the second player.
# Consider the strategy s<sub>2</sub>(''α'') of the second player.
#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),...}.
# Apply this strategy on an ω-game, generating (together with the second player's strategy ''s''<sub>2</sub>(''α'')) a sequence ⟨''a''<sub>1</sub>, ''b''<sub>2</sub>, ''a''<sub>3</sub>, ''b''<sub>4</sub>,&nbsp;..., a<sub>''t''</sub>, ''b''<sub>''t''+1</sub>,&nbsp;..., which does not belong to ''B.'' This is possible, because the number of choices for ⟨''a''<sub>1</sub>, ''a''<sub>3</sub>, ''a''<sub>5</sub>,&nbsp;... has the same cardinality as the continuum, which is larger than the cardinality of the proper initial portion {&nbsp;''β''&nbsp;∈&nbsp;''J''&nbsp;| ''β''&nbsp;≤&nbsp;''α''&nbsp;} of ''J.''
# Add this sequence to ''A'' to indicate that ''s''<sub>2</sub>(''α'') loses (on ⟨''a''<sub>1</sub>, ''a''<sub>3</sub>, ''a''<sub>5</sub>,&nbsp;...⟩).
#Decide that this sequence belongs to A, i.e. s2(T) lost.
# Process all possible strategies of ''S''<sub>1</sub> and ''S''<sub>2</sub> with [[transfinite induction]] on ''α.'' For all sequences that are not in ''A'' or ''B'' after that, decide arbitrarily whether they belong to ''A'' or to ''B,'' so that ''B'' is the complement of ''A.''
#Keep repeating with further strategies if there are any, making sure that sequences already considered do not become generated again. (We start from the set of all sequences and each time we generate a sequence and refute a strategy we project the generated sequence onto first player moves and onto second player moves, and we take away the two resulting sequences from our set of sequences.)
#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, prepare for an ω-game ''G''. For a given strategy ''s''<sub>1</sub> of the first player, there is an ''α''&nbsp;∈&nbsp;''J'' such that ''s''<sub>1</sub>&nbsp;=&nbsp;''s''<sub>1</sub>(''α''), and ''A'' has been constructed such that ''s''<sub>1</sub>(''α'') fails (on certain choices ⟨''b''<sub>2</sub>, ''b''<sub>4</sub>, ''b''<sub>6</sub>,&nbsp;...⟩ of the second player). Hence, ''s''<sub>1</sub> fails. Similarly, any other strategy of either player also fails.
Once this has been done we have a game ''G''. If you give me a strategy s1 then we considered that strategy at some time T = T(s1). At time ''T'', we decided an outcome of s1 that would be a loss of s1. Hence this strategy fails. But this is true for an arbitrary strategy; hence the axiom of determinacy and the axiom of choice are incompatible.


=== Using a choice function ===
== Infinitary logic and the axiom of determinacy ==
In this construction, the use of the axiom of choice is similar to the choice of socks as stated in the quote by Bertrand Russell at [[Axiom of choice#Quotations]].


In a ω-game, the two players are generating the sequence ⟨''a''<sub>1</sub>, ''b''<sub>2</sub>, ''a''<sub>3</sub>, ''b''<sub>4</sub>,&nbsp;...⟩, an element in ω<sup>ω</sup>, where our convention is that 0 is not a natural number, hence neither player can choose it. Define the function ''f'':&nbsp;ω<sup>ω</sup>&nbsp;→&nbsp;{0,&nbsp;1}<sup>ω</sup> such that ''f''(''r'') is the unique sequence of length ω with values are in {0, 1} whose first term equals 0, and whose sequence of runs (see [[run-length encoding]]) equals ''r.'' (Such an ''f'' can be shown to be injective. The image is the subset of {0,&nbsp;1}<sup>ω</sup> of sequences that start with 0 and that are not eventually constant. Formally, ''f'' is the [[Minkowski question mark function]], {0,&nbsp;1}<sup>ω</sup> is the [[Cantor space]] and ω<sup>ω</sup> is the [[Baire space (set theory)|Baire space]].)
Many different versions of [[infinitary logic]] were proposed in the late 20th century. One reason that has been given for believing in the axiom of determinacy is that it can be written as follows (in a version of infinite logic):


Observe the [[equivalence relation]] on {0,&nbsp;1}<sup>ω</sup> such that two sequences are equivalent if and only if they differ in a finite number of terms. This partitions the set into equivalence classes. Let ''T'' be the set of equivalence classes (such that ''T'' has the cardinality of the continuum). Define ''g'':&nbsp;{0,&nbsp;1}<sup>ω</sup>&nbsp;→&nbsp;''T'' that takes a sequence to its equivalence class. Define the complement of any sequence ''s'' in {0,&nbsp;1}<sup>ω</sup> to be the sequence s<sub>1</sub> that differs in each term. Define the function ''h'':&nbsp;''T''&nbsp;→&nbsp;''T'' such that for any sequence ''s'' in {0,&nbsp;1}<sup>ω</sup>, ''h'' applied to the equivalence class of ''s'' equals the equivalence class of the complement of ''s'' (which is well-defined because if ''s'' and ''s''' are equivalent, then their complements are equivalent). One can show that ''h'' is an [[Involution_(mathematics)|involution]] with no fixed points, and thus we have a partition of ''T'' into size-2 subsets such that each subset is of the form {''t'',&nbsp;''h''(''t'')}. Using the axiom of choice, we can choose one element out of each subset. In other words, we are choosing "half" of the elements of ''T,'' a subset that we denote by ''U'' (where U&nbsp;⊆&nbsp;T) such that ''t''&nbsp;∈&nbsp;''U'' iff ''h''(''t'')&nbsp;∉&nbsp;''U.''
<math>\forall G \subseteq Seq(S):</math>


Next, we define the subset ''A''&nbsp;⊆&nbsp;ω<sup>ω</sup> in which '''1''' wins: ''A'' is the set of all ''r'' such that ''g''(''f''(''r''))&nbsp;∈&nbsp;''U.'' We now claim that neither player has a winning strategy, using a [[strategy-stealing argument]]. Denote the current game state by a finite sequence of natural numbers (so that if the length of this sequence is even, then '''1''' is next to play; otherwise '''2''' is next to play).
<math>\forall a \in S: \exists a' \in S: \forall b \in S: \exists b' \in S: \forall c \in S: \exists c' \in S ... : (a,a',b,b',c,c'...) \in G </math> OR


Suppose that ''q'' is a (deterministic) winning strategy for '''2'''. Player '''1''' can construct a strategy ''p'' that beats ''q'' as follows: Suppose that player '''2'''{{`s}} response (according to ''q'') to ⟨1⟩ is ''b''<sub>1</sub>. Then '''1''' specifies in ''p'' that ''a''<sub>1</sub>&nbsp;=&nbsp;1&nbsp;+&nbsp;''b''<sub>1</sub>. (Roughly, '''1''' is now playing as '''2''' in a second parallel game; '''1'''{{`s}} winning set in the second game equals '''2'''{{`s}} winning set in the original game, and this is a contradiction. Nevertheless, we continue more formally.)
<math>\exists a \in S: \forall a' \in S: \exists b \in S: \forall b' \in S: \exists c \in S: \forall c' \in S ... :(a,a',b,b',c,c'...) \notin G </math>


Suppose that '''2'''{{`s}} response (always according to ''q'') to ⟨1&nbsp;+&nbsp;''b''<sub>1</sub>⟩ is ''b''<sub>2</sub>, and '''2'''{{`s}} response to ⟨1, ''b''<sub>1</sub>, ''b''<sub>2</sub>⟩ is ''b''<sub>3</sub>. We construct ''p'' for '''1''', we only aim to beat ''q,'' and therefore only have to handle the response ''b''<sub>2</sub> to '''1'''{{`s}} first move. Therefore set '''1'''{{`s}} response to ⟨1&nbsp;+&nbsp;''b''<sub>1</sub>, ''b''<sub>2</sub>⟩ is ''b''<sub>3</sub>. In general, for even ''n,'' denote '''2'''{{`s}} response to ⟨1&nbsp;+&nbsp;b<sub>1</sub>,&nbsp;..., b<sub>''n''−1</sub>⟩ by ''b''<sub>''n''</sub> and '''2'''{{`s}} response to ⟨1, ''b''<sub>1</sub>,&nbsp;..., ''b''<sub>''n''</sub>⟩ by ''b''<sub>''n''+1</sub>. Then '''1''' specify in ''p'' that '''1'''{{`s}} response to ⟨1&nbsp;+&nbsp;''b''<sub>1</sub>, ''b''<sub>2</sub>,&nbsp;..., ''b''<sub>''n''</sub>⟩ is ''b''<sub>''n''+1</sub>. Strategy ''q'' is presumed to be winning, and game-result ''r'' in ω<sup>ω</sup> given by ⟨1, ''b''<sub>1</sub>,&nbsp;...⟩ is one possible sequence allowed by ''q,'' so ''r'' must be winning for '''2''' and ''g''(''f''(''r'')) must not be in ''U.'' The game result ''r''<nowiki/>' in ω<sup>ω</sup> given by ⟨1&nbsp;+&nbsp;''b''<sub>1</sub>, ''b''<sub>2</sub>,&nbsp;...⟩ is also a sequence allowed by ''q'' (specifically, ''q'' playing against ''p''), so ''g''(''f''(''r''<nowiki/>')) must not be in ''U.'' However, ''f''(''r'') and ''f''(''r''<nowiki/>') differ in all but the first term (by the nature of run-length encoding and an offset of 1), so ''f''(''r'') and ''f''(''r''<nowiki/>') are in complement equivalent classes, so ''g''(''f''(''r'')), ''g''(''f''(''r''<nowiki/>')) cannot both be in ''U,'' contradicting the assumption that ''q'' is a winning strategy.
Note: Seq(''S'') is the set of all <math>\omega</math>-sequences of ''S''. The sentences here are infinitely long with a countably infinite list of [[Quantifiers (logic)|quantifiers]] where the ellipses appear.

Similarly, suppose that ''p'' is a winning strategy for '''1'''; the argument is similar but now uses the fact that equivalence classes were defined by allowing an arbitrarily large finite number of terms to differ. Let ''a''<sub>1</sub> be '''1'''{{`s}} first move. In general, for even ''n,'' denote '''1'''{{`s}} response to ⟨''a''<sub>1</sub>, 1⟩ (if ''n''&nbsp;=&nbsp;2) or ⟨''a''<sub>1</sub>, 1, ''a''<sub>2</sub>,&nbsp;..., ''a''<sub>n−1</sub>⟩ by ''a''<sub>''n''</sub> and '''1'''{{`s}} response to ⟨''a''<sub>1</sub>, 1&nbsp;+&nbsp;''a''<sub>2</sub>,&nbsp;... ''a''<sub>''n''</sub>⟩ by ''a''<sub>''n''+1</sub>. Then the game result ''r'' given by ⟨''a''<sub>1</sub>, 1, ''a''<sub>2</sub>, ''a''<sub>3</sub>,&nbsp;...⟩ is allowed by ''p'' so that ''g''(''f''(''r'')) must be in ''U''; also the game result ''r''<nowiki/>' given by ⟨''a''<sub>1</sub>, 1&nbsp;+&nbsp;''a''<sub>2</sub>, ''a''<sub>3</sub>,&nbsp;...⟩ is also allowed by ''p'' so that ''g''(''f''(''r''<nowiki/>')) must be in ''U.'' However, ''f''(''r'') and ''f''(''r''<nowiki/>') differ in all but the first ''a''<sub>1</sub>&nbsp;+&nbsp;1 terms, so they are in complement equivalent classes, therefore ''g''(''f''(''r'')) and ''g''(''f''(''r''<nowiki/>')) cannot both be in ''U,'' contradicting that ''p'' is a winning strategy.


== Large cardinals and the axiom of determinacy ==
== 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 cardinal]] axioms. By a theorem of [[W. Hugh Woodin|Woodin]], the consistency of [[Zermelo–Fraenkel set theory]] without choice (ZF) together with the axiom of determinacy is equivalent to the consistency of Zermelo–Fraenkel set theory with choice (ZFC) together with the existence of infinitely many [[Woodin cardinal]]s. Since Woodin cardinals are [[inaccessible cardinal|strongly inaccessible]], if AD is consistent, then so are an infinity of inaccessible cardinals.

The consistency of the axiom of determinacy is closely related to the question of the consistency of [[large cardinal]] axioms. By a theorem of [[W. Hugh Woodin|Woodin]], the consistency of Zermelo–Fraenkel set theory without choice (ZF) together with the axiom of determinacy is equivalent to the consistency of Zermelo–Fraenkel set theory with choice (ZFC) together with the existence of infinitely many [[Woodin cardinal]]s. Since Woodin cardinals are [[inaccessible cardinal|strongly inaccessible]], if AD is consistent, 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 cardinal]] larger than all of them, a very strong theory of [[Lebesgue measurable]] sets of reals emerges, as it is then provable that the axiom of determinacy is true in [[L(R)]], and therefore that ''every'' set of real numbers in L(R) is determined.
Moreover, if to the hypothesis of an infinite set of Woodin cardinals is added the existence of a [[measurable cardinal]] larger than all of them, a very strong theory of [[Lebesgue measurable]] sets of reals emerges, as it is then provable that the axiom of determinacy is true in [[L(R)]], and therefore that ''every'' set of real numbers in L(R) is determined.

== Projective ordinals ==
[[Yiannis Moschovakis]] introduced the ordinals δ{{su|b=''n''|p=1}}, which is the upper bound of the length of '''Δ'''{{su|b=''n''|p=1}}-norms (injections of a '''Δ'''{{su|b=''n''|p=1}} set into the ordinals), where '''Δ'''{{su|b=''n''|p=1}} is a level of the [[projective hierarchy]]. Assuming AD, all δ{{su|b=''n''|p=1}} are [[Initial ordinal|initial ordinals]], and we have {{nowrap|1=δ{{su|b=2''n''+2|p=1}} = (δ{{su|b=2''n''+1|p=1}})<sup>+</sup>}}, and for ''n''&nbsp;<&nbsp;ω, the {{nowrap|2''n''-th}} [[Suslin cardinal]] is equal to δ{{su|b=2''n''−1|p=1}}.<ref>V. G. Kanovei, [http://lab6.iitp.ru/en/pub/en_jms_1988_k.pdf The axiom of determinacy and the modern development of descriptive set theory], UDC 510.225; 510.223, Plenum Publishing Corporation (1988) p.270,282. Accessed 20 January 2023.</ref>


== See also ==
== See also ==

* [[Axiom of real determinacy]] (AD<sub>R</sub>)
* [[Axiom of real determinacy]] (AD<sub>R</sub>)
* [[Borel determinacy theorem]]
* [[Borel determinacy theorem]]
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==References==
==References==
{{refbegin}}
* {{Cite journal | last1=Mycielski | first1=Jan | author1-link = Jan Mycielski | last2=Steinhaus | first2=Hugo | author2-link = Hugo Steinhaus | title=A mathematical axiom contradicting the axiom of choice | mr=0140430 | year=1962 | journal=Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques | issn=0001-4117 | volume=10 | pages=1–3 }}
* {{Cite journal | last1=Mycielski | first1=Jan | author1-link = Jan Mycielski | last2=Steinhaus | first2=Hugo | author2-link = Hugo Steinhaus | title=A mathematical axiom contradicting the axiom of choice | mr=0140430 | year=1962 | journal=Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques | issn=0001-4117 | volume=10 | pages=1–3 }}
* {{cite journal|last1=Mycielski | first1=Jan | author1-link = Jan Mycielski | last2=Świerczkowski | first2=Stanisław | author2-link = Stanisław Świerczkowski|title=On the Lebesgue measurability and the axiom of determinateness|journal=Fund. Math.|volume=54|
* {{cite journal|last1=Mycielski | first1=Jan | author1-link = Jan Mycielski | last2=Świerczkowski | first2=Stanisław | author2-link = Stanisław Świerczkowski|title=On the Lebesgue measurability and the axiom of determinateness|journal=Fund. Math.|volume=54|
year=1964|pages=67–71}}
year=1964|pages=67–71| doi=10.4064/fm-54-1-67-71 |doi-access=free}}
* {{cite journal|author=Woodin, W. Hugh | author-link = W. Hugh Woodin |journal=[[Proceedings of the National Academy of Sciences of the United States of America]]|year=1988|title=Supercompact cardinals, sets of reals, and weakly homogeneous trees|volume=85|issue=18|pages=6587–6591|doi=10.1073/pnas.85.18.6587|pmid=16593979|pmc=282022}}
* {{cite journal|author=Woodin, W. Hugh | author-link = W. Hugh Woodin |journal=[[Proceedings of the National Academy of Sciences of the United States of America]]|year=1988|title=Supercompact cardinals, sets of reals, and weakly homogeneous trees|volume=85|issue=18|pages=6587–6591|doi=10.1073/pnas.85.18.6587|pmid=16593979|pmc=282022| bibcode = 1988PNAS...85.6587W | doi-access = free }}
* {{cite journal|last1=Martin |first1=Donald A. |author1-link=Donald A. Martin |first2=John R. |last2=Steel |author2-link=John R. Steel |date=Jan 1989|title=A Proof of Projective Determinacy |journal=[[Journal of the American Mathematical Society]] |volume=2 |issue=1 |pages=71–125 |doi=10.2307/1990913 |url=https://www.ams.org/journals/jams/1989-02-01/S0894-0347-1989-0955605-X/S0894-0347-1989-0955605-X.pdf |url-status=dead |archive-url=https://web.archive.org/web/20160430183734/http://www.ams.org/journals/jams/1989-02-01/S0894-0347-1989-0955605-X/S0894-0347-1989-0955605-X.pdf |archive-date=April 30, 2016 |jstor=1990913 |doi-access=free }}
* {{cite journal|last1=Martin |first1=Donald A. |author1-link=Donald A. Martin |first2=John R. |last2=Steel |author2-link=John R. Steel |date=Jan 1989|title=A Proof of Projective Determinacy |journal=[[Journal of the American Mathematical Society]] |volume=2 |issue=1 |pages=71–125 |doi=10.2307/1990913 |jstor=1990913 |doi-access=free }}
* {{cite book|last=Jech | first = Thomas | author-link = Thomas Jech | title=Set theory, third millennium edition (revised and expanded)|publisher=Springer|year=2002|isbn=978-3-540-44085-7}}
* {{cite book|last=Jech | first = Thomas | author-link = Thomas Jech | title=Set theory, third millennium edition (revised and expanded)|publisher=Springer|year=2002|isbn=978-3-540-44085-7}}
* {{cite book|last=Kanamori | first = Akihiro|author-link=Akihiro Kanamori|title=The Higher Infinite|title-link=The Higher Infinite|edition=2nd|year=2008|publisher=Springer Science & Business Media|isbn=978-3-540-88866-6}}
* {{cite book|last=Kanamori | first = Akihiro|author-link=Akihiro Kanamori|title=The Higher Infinite|title-link=The Higher Infinite|edition=2nd|year=2008|publisher=Springer Science & Business Media|isbn=978-3-540-88866-6}}
* {{cite book|last1=Moschovakis |first1=Yiannis N. |author-link1=Yiannis N. Moschovakis |title=Descriptive set theory |date=2009 |publisher=American Mathematical Society |location=Providence, R.I. |isbn=978-0-8218-4813-5 |edition=2nd |url=http://www.math.ucla.edu/~ynm/lectures/dst2009/dst2009.pdf |url-status=bot: unknown |archive-url=https://web.archive.org/web/20141112111558/http://www.math.ucla.edu/~ynm/lectures/dst2009/dst2009.pdf |archive-date=2014-11-12 }}
* {{cite book|last1=Moschovakis |first1=Yiannis N. |author-link1=Yiannis N. Moschovakis |title=Descriptive set theory |date=2009 |publisher=American Mathematical Society |location=Providence, R.I. |isbn=978-0-8218-4813-5 |edition=2nd |url=https://www.math.ucla.edu/~ynm/lectures/dst2009/dst2009.pdf |url-status=live|archive-url=https://web.archive.org/web/20141112111558/http://www.math.ucla.edu/~ynm/lectures/dst2009/dst2009.pdf |archive-date=2014-11-12 }}
{{refend}}


===Inline citations===
== Further reading ==
{{Reflist}}


== Further reading ==
* Philipp Rohde, ''On Extensions of the Axiom of Determinacy'', Thesis, Department of Mathematics, University of Bonn, Germany, 2001
* Philipp Rohde, ''On Extensions of the Axiom of Determinacy'', Thesis, Department of Mathematics, University of Bonn, Germany, 2001
* [[Rastislav J. Telgársky|Telgársky, R.J.]] [http://telgarska.com/1987-RMJM-Telgarsky-Topological-Games.pdf ''Topological Games: On the 50th Anniversary of the Banach-Mazur Game''], Rocky Mountain J. Math. '''17''' (1987), pp.&nbsp;227–276. (3.19 MB)
* [[Rastislav J. Telgársky|Telgársky, R.J.]] [http://telgarska.com/1987-RMJM-Telgarsky-Topological-Games.pdf ''Topological Games: On the 50th Anniversary of the Banach-Mazur Game''], Rocky Mountain J. Math. '''17''' (1987), pp.&nbsp;227–276. (3.19 MB)

Latest revision as of 18:39, 24 November 2024

In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of a certain type is determined; that is, one of the two players has a winning strategy.

Steinhaus and Mycielski's motivation for AD was its interesting consequences, and suggested that AD could be true in the smallest natural model L(R) of a set theory, which accepts only a weak form of the axiom of choice (AC) but contains all real and all ordinal numbers. Some consequences of AD followed from theorems proved earlier by Stefan Banach and Stanisław Mazur, and Morton Davis. Mycielski and Stanisław Świerczkowski contributed another one: AD implies that all sets of real numbers are Lebesgue measurable. Later Donald A. Martin and others proved more important consequences, especially in descriptive set theory. In 1988, John R. Steel and W. Hugh Woodin concluded a long line of research. Assuming the existence of some uncountable cardinal numbers analogous to ℵ0, they proved the original conjecture of Mycielski and Steinhaus that AD is true in L(R).

Types of game that are determined

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The axiom of determinacy refers to games of the following specific form: Consider a subset A of the Baire space ωω of all infinite sequences of natural numbers. Two players alternately pick natural numbers

n0, n1, n2, n3, ...

That generates the sequence ⟨nii∈ω after infinitely many moves. The player who picks first wins the game if and only if the sequence generated is an element of A. The axiom of determinacy is the statement that all such games are determined.

Not all games require the axiom of determinacy to prove them determined. If the set A is clopen, the game is essentially a finite game, and is therefore determined. Similarly, if A is a closed set, then the game is determined. By the Borel determinacy theorem, games whose winning set is a Borel set are determined. It follows from the existence of sufficiently large cardinals that AD holds in L(R) and that a game is determined if it has a projective set as its winning set (see Projective determinacy).

The axiom of determinacy implies that for every subspace X of the real numbers, the Banach–Mazur game BM(X) is determined, and consequently, that every set of reals has the property of Baire.

Incompatibility with the axiom of choice

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Under assumption of the axiom of choice, we present two separate constructions of counterexamples to the axiom of determinacy. It follows that the axiom of determinacy and the axiom of choice are incompatible.

Using a well-ordering of the continuum

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The set S1 of all first player strategies in an ω-game G has the same cardinality as the continuum. The same is true for the set S2 of all second player strategies. Let SG be the set of all possible sequences in G, and A be the subset of sequences of SG that make the first player win. With the axiom of choice we can well order the continuum, and we can do so in such a way that any proper initial portion has lower cardinality than the continuum. We use the obtained well ordered set J to index both S1 and S2, and construct A such that it will be a counterexample.

We start with empty sets A and B. Let α ∈ J be the index of the strategies in S1 and S2. We need to consider all strategies S1 = {s1(α)}αJ of the first player and all strategies S2 = {s2(α)}αJ of the second player to make sure that for every strategy there is a strategy of the other player that wins against it. For every strategy of the player considered we will generate a sequence that gives the other player a win. Let t be the time whose axis has length ℵ0 and which is used during each game sequence. We create the counterexample A by transfinite recursion on α:

  1. Consider the strategy s1(α) of the first player.
  2. Apply this strategy on an ω-game, generating (together with the first player's strategy s1(α)) a sequence ⟨a1, b2, a3, b4, ...,at, bt+1, ...⟩, which does not belong to A. This is possible, because the number of choices for ⟨b2, b4, b6, ...⟩ has the same cardinality as the continuum, which is larger than the cardinality of the proper initial portion { β ∈ J | β < α } of J.
  3. Add this sequence to B to indicate that s1(α) loses (on ⟨b2, b4, b6, ...⟩).
  4. Consider the strategy s2(α) of the second player.
  5. Apply this strategy on an ω-game, generating (together with the second player's strategy s2(α)) a sequence ⟨a1, b2, a3, b4, ..., at, bt+1, ...⟩, which does not belong to B. This is possible, because the number of choices for ⟨a1, a3, a5, ...⟩ has the same cardinality as the continuum, which is larger than the cardinality of the proper initial portion { β ∈ J | β ≤ α } of J.
  6. Add this sequence to A to indicate that s2(α) loses (on ⟨a1, a3, a5, ...⟩).
  7. Process all possible strategies of S1 and S2 with transfinite induction on α. For all sequences that are not in A or B after that, decide arbitrarily whether they belong to A or to B, so that B is the complement of A.

Once this has been done, prepare for an ω-game G. For a given strategy s1 of the first player, there is an α ∈ J such that s1 = s1(α), and A has been constructed such that s1(α) fails (on certain choices ⟨b2, b4, b6, ...⟩ of the second player). Hence, s1 fails. Similarly, any other strategy of either player also fails.

Using a choice function

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In this construction, the use of the axiom of choice is similar to the choice of socks as stated in the quote by Bertrand Russell at Axiom of choice#Quotations.

In a ω-game, the two players are generating the sequence ⟨a1, b2, a3, b4, ...⟩, an element in ωω, where our convention is that 0 is not a natural number, hence neither player can choose it. Define the function f: ωω → {0, 1}ω such that f(r) is the unique sequence of length ω with values are in {0, 1} whose first term equals 0, and whose sequence of runs (see run-length encoding) equals r. (Such an f can be shown to be injective. The image is the subset of {0, 1}ω of sequences that start with 0 and that are not eventually constant. Formally, f is the Minkowski question mark function, {0, 1}ω is the Cantor space and ωω is the Baire space.)

Observe the equivalence relation on {0, 1}ω such that two sequences are equivalent if and only if they differ in a finite number of terms. This partitions the set into equivalence classes. Let T be the set of equivalence classes (such that T has the cardinality of the continuum). Define g: {0, 1}ω → T that takes a sequence to its equivalence class. Define the complement of any sequence s in {0, 1}ω to be the sequence s1 that differs in each term. Define the function hT → T such that for any sequence s in {0, 1}ω, h applied to the equivalence class of s equals the equivalence class of the complement of s (which is well-defined because if s and s' are equivalent, then their complements are equivalent). One can show that h is an involution with no fixed points, and thus we have a partition of T into size-2 subsets such that each subset is of the form {th(t)}. Using the axiom of choice, we can choose one element out of each subset. In other words, we are choosing "half" of the elements of T, a subset that we denote by U (where U ⊆ T) such that t ∈ U iff h(t) ∉ U.

Next, we define the subset A ⊆ ωω in which 1 wins: A is the set of all r such that g(f(r)) ∈ U. We now claim that neither player has a winning strategy, using a strategy-stealing argument. Denote the current game state by a finite sequence of natural numbers (so that if the length of this sequence is even, then 1 is next to play; otherwise 2 is next to play).

Suppose that q is a (deterministic) winning strategy for 2. Player 1 can construct a strategy p that beats q as follows: Suppose that player 2's response (according to q) to ⟨1⟩ is b1. Then 1 specifies in p that a1 = 1 + b1. (Roughly, 1 is now playing as 2 in a second parallel game; 1's winning set in the second game equals 2's winning set in the original game, and this is a contradiction. Nevertheless, we continue more formally.)

Suppose that 2's response (always according to q) to ⟨1 + b1⟩ is b2, and 2's response to ⟨1, b1, b2⟩ is b3. We construct p for 1, we only aim to beat q, and therefore only have to handle the response b2 to 1's first move. Therefore set 1's response to ⟨1 + b1, b2⟩ is b3. In general, for even n, denote 2's response to ⟨1 + b1, ..., bn−1⟩ by bn and 2's response to ⟨1, b1, ..., bn⟩ by bn+1. Then 1 specify in p that 1's response to ⟨1 + b1, b2, ..., bn⟩ is bn+1. Strategy q is presumed to be winning, and game-result r in ωω given by ⟨1, b1, ...⟩ is one possible sequence allowed by q, so r must be winning for 2 and g(f(r)) must not be in U. The game result r' in ωω given by ⟨1 + b1, b2, ...⟩ is also a sequence allowed by q (specifically, q playing against p), so g(f(r')) must not be in U. However, f(r) and f(r') differ in all but the first term (by the nature of run-length encoding and an offset of 1), so f(r) and f(r') are in complement equivalent classes, so g(f(r)), g(f(r')) cannot both be in U, contradicting the assumption that q is a winning strategy.

Similarly, suppose that p is a winning strategy for 1; the argument is similar but now uses the fact that equivalence classes were defined by allowing an arbitrarily large finite number of terms to differ. Let a1 be 1's first move. In general, for even n, denote 1's response to ⟨a1, 1⟩ (if n = 2) or ⟨a1, 1, a2, ..., an−1⟩ by an and 1's response to ⟨a1, 1 + a2, ... an⟩ by an+1. Then the game result r given by ⟨a1, 1, a2, a3, ...⟩ is allowed by p so that g(f(r)) must be in U; also the game result r' given by ⟨a1, 1 + a2, a3, ...⟩ is also allowed by p so that g(f(r')) must be in U. However, f(r) and f(r') differ in all but the first a1 + 1 terms, so they are in complement equivalent classes, therefore g(f(r)) and g(f(r')) cannot both be in U, contradicting that p is a winning strategy.

Large cardinals and the axiom of determinacy

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The consistency of the axiom of determinacy is closely related to the question of the consistency of large cardinal axioms. By a theorem of Woodin, the consistency of Zermelo–Fraenkel set theory without choice (ZF) together with the axiom of determinacy is equivalent to the consistency of Zermelo–Fraenkel set theory with choice (ZFC) together with the existence of infinitely many Woodin cardinals. Since Woodin cardinals are strongly inaccessible, if AD is consistent, 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 cardinal larger than all of them, a very strong theory of Lebesgue measurable sets of reals emerges, as it is then provable that the axiom of determinacy is true in L(R), and therefore that every set of real numbers in L(R) is determined.

Projective ordinals

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Yiannis Moschovakis introduced the ordinals δ1
n
, which is the upper bound of the length of Δ1
n
-norms (injections of a Δ1
n
set into the ordinals), where Δ1
n
is a level of the projective hierarchy. Assuming AD, all δ1
n
are initial ordinals, and we have δ1
2n+2
= (δ1
2n+1
)+
, and for n < ω, the 2n-th Suslin cardinal is equal to δ1
2n−1
.[1]

See also

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References

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Inline citations

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  1. ^ V. G. Kanovei, The axiom of determinacy and the modern development of descriptive set theory, UDC 510.225; 510.223, Plenum Publishing Corporation (1988) p.270,282. Accessed 20 January 2023.

Further reading

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