The Strength Turing Determinacy within Second Order Arithmetic · 2015-08-09 · 2: Second Order Arithmetic = First Order Arithmetic plus set variables X (and quanti–ers) with the

The Strength Turing Determinacy within Second OrderArithmetic

Richard A. Shore Cornell University

Logic Coloquium 15Helsinki, FinlandAugust 5, 2015

Richard A. Shore () Turing Determinacy 8/5/2015 1 / 33

References

References

This is joint work with Antonio Montalbán.

[MS15] The Strength of Turing Determinacy within Second OrderArithmetic, Fundamenta Mathematica, to appear.

Our previous papers on ordinary determinacy in second order arithmetic:[MS12] The limits of determinacy in second order arithmetic, Proceedingsof the London Mathematical Society 104 (3) (2012), 223-252.

[MS14] The Limits of Determinacy in Second Order Arithmetic:Consistency and Complexity Strength, Israel Journal of Mathematics 204(2014), 477-508.

Richard A. Shore (Cornell University ) Turing Determinacy 8/5/2015 2 / 33

The Setting

The Setting

Measuring the Strength of Theorems and Systems

Recursion/Set Theoretic: Higher Turing degree or De�nitionalComplexity of Solutions/Size of Ordinals or levels of Admissibility ininitial segments of L.

Reverse Mathematical: Logical implication: "stronger" theorem(theory) proves "weaker" one over a weak base theory (RCA0).

Consistency Strength: Prove the consistency of the �weaker� theorem(theory) from the "stronger".

We work in Z2: Second Order Arithmetic = First Order Arithmetic plusset variables X (and quanti�ers) with the membership relation n 2 X .

Five standard subsystems in reverse mathematics.


The Setting

The Setting








The Setting

The Setting








The Setting

The Setting








The Setting

The Setting






Five standard subsystems in reverse mathematics.Richard A. Shore (Cornell University ) Turing Determinacy 8/5/2015 3 / 33

The Setting

Common Systems

Each system of reverse mathematics contains basic axioms for +, �, < and(I0): 0 2 X ^ 8n (n 2 X ! n + 1 2 X ))! 8n (n 2 X ).

1 (RCA0): for '; 2 �01:

(�01-CA0) 8n ('(n)$ : (n))! 9X 8n (n 2 X $ '(n));(I�1) ('(0) ^ 8n ('(n)! '(n + 1)))! 8n'(n).

2 (WKL0) RCA0 + every in�nite subtree of 2<! has an in�nite path.3 (ACA0) RCA0 + 9X 8n (n 2 X $ '(n)), ' arithmetic.4 (ATR0) RCA0 + If <X is a well order <X with domain D andh'x (z ;Z )jx 2 Di are arithmetic, then 9hKx j x 2 Di(if y is theimmediate successor of x , then 8n (n 2 Ky $ 'x (n;Kx )), and if x isa limit point, then Kx =

LfKy j y <X xg).

5 (�11-CA0) 9X 8k (k 2 X $ '(k)) for ' 2 �11.


The Setting

Common Systems


1 (RCA0): for '; 2 �01:



LfKy j y <X xg).

5 (�11-CA0) 9X 8k (k 2 X $ '(k)) for ' 2 �11.


The Setting

Common Systems


1 (RCA0): for '; 2 �01:



LfKy j y <X xg).

5 (�11-CA0) 9X 8k (k 2 X $ '(k)) for ' 2 �11.


The Setting

Theme

In each setting we examine theorems (usually existence and soconstructions). Our underlying theme is determining how hard it is toconstruct sets or functions asserted to exist or to prove their existence.

What information, e.g. some number of jumps, is needed to carry out orverify the construction? What axioms in second order arithmetic arerequired to prove that the construction works? How do these relate to thelogical (i.e. reverse mathematical) or proof theoretic (i.e. consistency)strength of other theorems or systems?

Each approach reveals information about the �e¤ectiveness�of theconstruction. Sometimes basically the same information. Sometimes �neror di¤erent.These systems often called the Big Five. For years almost all theoremsstudied were equivalent to one of them.


The Setting

Theme





The Setting

Theme



Each approach reveals information about the �e¤ectiveness�of theconstruction. Sometimes basically the same information. Sometimes �neror di¤erent.

These systems often called the Big Five. For years almost all theoremsstudied were equivalent to one of them.


The Setting

Theme





The Setting

Uncommon Systems

Although, percentage-wise, they still represent relatively few classicaltheorems and subjects, there are now many systems (a zoo) that are not inthis short hierarchy but lie well below ACA0 and are either weaker than orincomparable to WKL0.

Our previous work on determinacy added a couple but also went farbeyond all these systems climbing up the comprehension hierarchy all theway to full second order arithmetic.

(�1n-CA0) 9X 8k (k 2 X $ '(k)) for ' 2 �1n.(�1n-CA0) 9X 8k (k 2 X $ '(k)) for ' 2 �1n for which there is anequivalent 2 �1n.

The union of either of these interleaved hierarchies constitute Z2, fullsecond order arithmetic.There are one or two known mathematical theorems at the level of�12-CA0 and nothing above �

13-CA0. We provided a series going all the

way up these hierarchies to Z2 and beyond.


The Setting

Uncommon Systems




The union of either of these interleaved hierarchies constitute Z2, fullsecond order arithmetic.

There are one or two known mathematical theorems at the level of�12-CA0 and nothing above �


way up these hierarchies to Z2 and beyond.


The Setting

Uncommon Systems




The union of either of these interleaved hierarchies constitute Z2, fullsecond order arithmetic.There are one or two known mathematical theorems at the level of�12-CA0 and nothing above �


way up these hierarchies to Z2 and beyond.Richard A. Shore (Cornell University ) Turing Determinacy 8/5/2015 6 / 33

The Setting

Conservativity and ZFC�

To set a global connection between second order arithmetic and set theorywe point to ZFC�, ZFC without the power set axiom and note that for ourpurposes it is of the same strength as Z2.

De�nitionIf � is a class of formulas, then a theory T is �-conservative over a theoryS if every theorem of T which is a sentence in � is also a theorem of S .(Usually T extends S .)

Proposition

ZFC� (even with a de�nable well ordering of the universe, e.g.ZFC� + V = L) is �14-conservative over Z2.

Idea: If start withM � Z2 with a counterexample X to the �14 sentence,take a coding of L[X ] �ZFC� inM. Argue by absoluteness that Xremains a counterexample. Should be "well known".Note (Feferman and Levy 60�s): ZFC� is not �15-conservative over Z2.


The Setting




Proposition


Idea: If start withM � Z2 with a counterexample X to the �14 sentence,take a coding of L[X ] �ZFC� inM. Argue by absoluteness that Xremains a counterexample. Should be "well known".Note (Feferman and Levy 60�s): ZFC� is not �15-conservative over Z2.


The Setting




Proposition


Idea: If start withM � Z2 with a counterexample X to the �14 sentence,take a coding of L[X ] �ZFC� inM. Argue by absoluteness that Xremains a counterexample. Should be "well known".Note (Feferman and Levy 60�s): ZFC� is not �15-conservative over Z2.Richard A. Shore (Cornell University ) Turing Determinacy 8/5/2015 7 / 33

Determinacy and Reverse Mathematics

Games and Determinacy

De�nitionOur games are played by two players I and II. They alternate playing 0 or1 with I playing �rst to produce a play of the game which is a sequencex 2 2!. A game GA is speci�ed by a subset A of 2!. We say that I wins aplay x of the game GA speci�ed by A if x 2 A. Otherwise II wins thatplay. A strategy for I (II) is a function � from binary strings p of even(odd) length into f0; 1g. It is a wining strategy if any play x following it(i.e. x(n) = �(x � n) for every even (odd) n) is a win for for I (II). We saythat the game GA is determined if there is a winning strategy for I or II inthis game. If � is a class of sets A, then we say that � is determined if GAis determined for every A 2 �. We denote the assertion that � isdetermined by � determinacy or �-DET.



Early Results

Theme: One hopes that, for �simple��, �-DET will be provable in�simple� systems and have �simple� strategies.

1 Early work (50�s) produced �elementary� combinatorial proofs up to�03.

2 Friedman 1971: Borel-DET needs @1 many iterations of power set.�05-DET is not provable in Z2.

3 Martin 1974: �04-DET not provable in Z2.4 In RCA0: Steel [1976] �01-DET$ATR0; Tanaka [1991]�01 ^ �01-DET$ �11-CA0; work on levels up to �

03 (Tanaka [1991] et

al. [2007]; Welch [2009]).

Martin [1974] also sketched a proof that �04-DET is provable in Z2.However, we discovered that Martin�s proof of �04-DET does not work inZ2. Instead we found a hierarchy of classes well within �04 for whichdeterminacy is provable precisely at each �1n.



Early Results






al. [2007]; Welch [2009]).




Early Results




3 Martin 1974: �04-DET not provable in Z2.

4 In RCA0: Steel [1976] �01-DET$ATR0; Tanaka [1991]�01 ^ �01-DET$ �11-CA0; work on levels up to �


al. [2007]; Welch [2009]).




Early Results






al. [2007]; Welch [2009]).




Early Results






al. [2007]; Welch [2009]).

Martin [1974] also sketched a proof that �04-DET is provable in Z2.

However, we discovered that Martin�s proof of �04-DET does not work inZ2. Instead we found a hierarchy of classes well within �04 for whichdeterminacy is provable precisely at each �1n.



Early Results






al. [2007]; Welch [2009]).




Reverse Mathematics: Montalbán and Shore [2012]

De�nition (The Di¤erence Hierarchy on �03 sets)

The n-�03 sets are those of the form (A0 � A1) [ (A2 � A3) [ (A4 � : : :Anfor Ai 2 �03. The !-�03 sets, [fn-�03jn 2 !g, are the closure of the �03 setsunder �nite Boolean combinations.

Theorem (For n � 1, (Welch for n = 1))

�1n+2-CA0 ` n-�03-DET.

Idea: A careful implementation of Martin�s proof. Uses the�14-conservativity of �

1n+2-DC0 over �

1n+2-CA0.

Theorem

�1n+2-CA 0 n-�03-DET.







�1n+2-CA0 ` n-�03-DET.


1n+2-DC0 over �

1n+2-CA0.

Theorem

�1n+2-CA 0 n-�03-DET.







�1n+2-CA0 ` n-�03-DET.


1n+2-DC0 over �

1n+2-CA0.

Theorem

�1n+2-CA 0 n-�03-DET.







�1n+2-CA0 ` n-�03-DET.


1n+2-DC0 over �

1n+2-CA0.

Theorem

�1n+2-CA 0 n-�03-DET.



So even !-�03-DET is beyond second order arithmetic by compactness.

Corollary

Z2 0 !-�03�DET and so a fortiori, Z2 0 �04-DET.

The natural question from the viewpoint of reverse mathematics now iscan we reverse these implications (over some reasonable base theory).However,

No Reversals:

Theorem!-�03-DET0 �12-CA0 and !-�03-DET + �1n+2-CA0 0 �1n+2-CA0.

Idea: Skolem-Löwenheim arguments in fragments of L preservedeterminacy but not admissibility (under unions). So comprehension fails.Thus !- �03-DET and Z2 are reverse mathematically incomparable.




Corollary



No Reversals:






Corollary



No Reversals:






Corollary



No Reversals:




Determinacy and Consistency Consistency

Consistency Strength: Montalbán and Shore [2014]

Question (John Steel)

Does !-�03-DET prove the consistency of Z2.

This seemed like a question for a proof theorist as the above results showthat the existence of even an !-model of Z2 proves the consistency of!-�03-DET. So !-�

03-DET cannot prove the existence of an !-model of

Z2. How does a recursion theorist construct a non !-model? Carefully.

More generally we can ask if we can restore the linearity of these sequencesof systems by moving to consistency rather than logical strength?Indeed, we can and the heart of our proof is actually recursion/settheoretic. We uniformly analyze the strength of the steps along the way,n-�03-DET, in terms of analogs to the jump operators used at lower levels.








Z2. How does a recursion theorist construct a non !-model? Carefully.More generally we can ask if we can restore the linearity of these sequencesof systems by moving to consistency rather than logical strength?

Indeed, we can and the heart of our proof is actually recursion/settheoretic. We uniformly analyze the strength of the steps along the way,n-�03-DET, in terms of analogs to the jump operators used at lower levels.








Z2. How does a recursion theorist construct a non !-model? Carefully.More generally we can ask if we can restore the linearity of these sequencesof systems by moving to consistency rather than logical strength?Indeed, we can and the heart of our proof is actually recursion/settheoretic. We uniformly analyze the strength of the steps along the way,n-�03-DET, in terms of analogs to the jump operators used at lower levels.


Determinacy and Consistency Recursion (Jump) Theoretic Measures

The Recursion/Set Theoretic Result

Main Technical recursion/set theoretic result:

Theorem (Main Theorem)

For n � 1, n-�03-DET ` 8X (�Xn+1 exists. (�Xn+1 is the least �n admissibleordinal containing X .) (n = 1 is due to Welch).

To express our proof theoretic results we de�ne a partial order T >c S andan operator which is strictly increasing in this order.

De�nitionIf T and S are theories in the language of second order arithmetic, thenT >c S if T ` Con(S). We let �(T ) be the theory which says that forevery set X there is a �-model of T containing X :

Not only does �(T ) ` Con(T ) but it is signi�cantly stronger in terms ofconsistency strength. Indeed, even the existence of an !-model of Tproves Con(T ); Con(T + Con(T )); etc.


Determinacy and Consistency Recursion (Jump) Theoretic Measures

The Recursion/Set Theoretic Result

Main Technical recursion/set theoretic result:

Theorem (Main Theorem)

For n � 1, n-�03-DET ` 8X (�Xn+1 exists. (�Xn+1 is the least �n admissibleordinal containing X .) (n = 1 is due to Welch).

To express our proof theoretic results we de�ne a partial order T >c S andan operator which is strictly increasing in this order.

De�nitionIf T and S are theories in the language of second order arithmetic, thenT >c S if T ` Con(S). We let �(T ) be the theory which says that forevery set X there is a �-model of T containing X :

Not only does �(T ) ` Con(T ) but it is signi�cantly stronger in terms ofconsistency strength. Indeed, even the existence of an !-model of Tproves Con(T ); Con(T + Con(T )); etc.Richard A. Shore (Cornell University ) Turing Determinacy 8/5/2015 13 / 33

Determinacy and Consistency Reverse Mathematical Results

Reverse Mathematical Results

Reverse mathematically, our results are as follows:

Theorem

For every n � 1 we have the following provability relations none of whichcan be reversed.

1 �1n+2-CA0 ` n-�03-DET.2 �1n+2-CA0 ` �(n-�03-DET).3 �(�1n+2-CA0) ` �(n-�03-DET).4 �(n- �03-DET) ` n-�03-DET.5 n-�03-DET` �(�1n+2-CA0).6 �(n-�03-DET) ` �(�1n+2-CA0).

Here it is (5) that is the main new recursion/set theoretic result of[MS14]. Most of the others are essentially in [MS12].


Determinacy and Consistency Reverse Mathematical Results

Diagram of above list of results:

�(�1n+2-CA0)+

�1n+2-CA0 =) �(n-�03-DET)+ +. n-�03-DET. +. �(�1n+2-CA0)+ +

�1n+2-CA0 =) �(�1n+1-CA0)+

�1n+1-CA0


Determinacy and Consistency Corollaries

Corollaries: Reverse Math and Recursion/Set Theory

Corollary (of the proofs plus Skloem-Löwenheim)

For every X , the least � such that L�[X ] � n-�03-DET is a limit ofadmissible ordinals strictly between ��n[X ], the �rst limit of �n-admissiblescontaining X , and �n[X ], the least �n-nonprojectable containing X .


The least ordinal � such that L� � !-�03-DET is [�n, the supremum of theleast �n admissibles over n 2 !. It is also the least such that L� containswinning strategies for all light-faced !-�03 games. This ordinal is far below�0, the least � such that L� � Z2.

Corollary (the following are equivalent over RCA0:)

(1) !-�03-DET and (2) 8n(�(�1n-CA0)).









(1) !-�03-DET and (2) 8n(�(�1n-CA0)).









(1) !-�03-DET and (2) 8n(�(�1n-CA0)).



Proof Theoretic Corollaries

Corollary (Restoring Linearity)For every n � 1 we have the following chain of consistency strengthrelations:� � ��1n+1-CA0 <c �

1n+2-CA0 <c n-�03-DET <c �

1n+2-CA0 � � �

Corollary (Equiconsistency)The following theories are equiconsistent:

1 Z2, i.e., the scheme which contains, for each n 2 N, the axiom�1n-CA0.

2 The scheme which contains, for each n 2 N, the axiom n-�03-DET.3 ZFC�, i.e. ZFC without the power set axiom.




Corollary (Restoring Linearity)For every n � 1 we have the following chain of consistency strengthrelations:� � ��1n+1-CA0 <c �

1n+2-CA0 <c n-�03-DET <c �

1n+2-CA0 � � �

Corollary (Equiconsistency)The following theories are equiconsistent:

1 Z2, i.e., the scheme which contains, for each n 2 N, the axiom�1n-CA0.

2 The scheme which contains, for each n 2 N, the axiom n-�03-DET.3 ZFC�, i.e. ZFC without the power set axiom.




Corollary

In contrast, !-�03-DET >c Z2 and indeedZ2 <c Z2 + Con(Z2) <c Z2 + Con(Z2) + Con(Z2 + Con(Z2)) <c � � �

� � � <c !-�03-DET.However, !- �03-DET does not prove that there is an !-model of Z2 asthat would imply the consistency of !- �03-DET.These relations also hold between !-�03-DET and ZFC

� in place of Z2.

Corollary

ZFC� (and so also Z2) is a �13 conservative extension ofRCA0 + fn-�03-DET jn 2 !g.




Corollary

In contrast, !-�03-DET >c Z2 and indeedZ2 <c Z2 + Con(Z2) <c Z2 + Con(Z2) + Con(Z2 + Con(Z2)) <c � � �

� � � <c !-�03-DET.However, !- �03-DET does not prove that there is an !-model of Z2 asthat would imply the consistency of !- �03-DET.These relations also hold between !-�03-DET and ZFC

� in place of Z2.

Corollary

ZFC� (and so also Z2) is a �13 conservative extension ofRCA0 + fn-�03-DET jn 2 !g.


Determinacy and Turing Determinacy

Turing Determinacy: Montalbán and Shore [2015]

De�nitionAn A � 2![!!] is Turing invariant or degree closed if(8f 2 2![!!])(8g 2 2![!!])(f �T g ! (f 2 A$ g 2 A)). We denote by� Turing determinacy or �-TD the assertion that every degree closedA 2 � is determined.

Theorem (Martin)A degree closed set A is determined if and only if it contains a cone, i.e. aset of Turing degrees of the form fxjx � zg for some degree z called thebase of the cone or is disjoint from a cone.

There are some important relations between Determinacy and TuringDeterminacy at high levels up to eventual agreement.



Turing Determinacy: Montalbán and Shore [2015]

De�nitionAn A � 2![!!] is Turing invariant or degree closed if(8f 2 2![!!])(8g 2 2![!!])(f �T g ! (f 2 A$ g 2 A)). We denote by� Turing determinacy or �-TD the assertion that every degree closedA 2 � is determined.

Theorem (Martin)A degree closed set A is determined if and only if it contains a cone, i.e. aset of Turing degrees of the form fxjx � zg for some degree z called thebase of the cone or is disjoint from a cone.

There are some important relations between Determinacy and TuringDeterminacy at high levels up to eventual agreement.



Theorem (Martin [70]; Harrington [78])

�11-DET is equivalent to �11-TD.

Theorem (Friedman 71; Martin 1970�s)Borel-DET is equivalent to Borel-TD.

Theorem (Martin improving Friedman 71)

�0�-DET implies �0�+1-TD which implies ��+2-TD while �

0�+5-TD

implies �0�+4-DET.

Theorem (Woodin)Full Determinacy and Turing Determinacy are equiconsistent andequivalent (over DC) in L(R).

Di¤erences appear at low levels. There are a few results from the 70s byHarrington-Kechris and by Martin. We translate their results into thesetting of reverse mathematics, re�ne a few and �ll in some of the gaps.



Theorem (Martin [70]; Harrington [78])

�11-DET is equivalent to �11-TD.

Theorem (Friedman 71; Martin 1970�s)Borel-DET is equivalent to Borel-TD.

Theorem (Martin improving Friedman 71)

�0�-DET implies �0�+1-TD which implies ��+2-TD while �

0�+5-TD

implies �0�+4-DET.

Theorem (Woodin)Full Determinacy and Turing Determinacy are equiconsistent andequivalent (over DC) in L(R).

Di¤erences appear at low levels. There are a few results from the 70s byHarrington-Kechris and by Martin. We translate their results into thesetting of reverse mathematics, re�ne a few and �ll in some of the gaps.Richard A. Shore (Cornell University ) Turing Determinacy 8/5/2015 20 / 33

Turing Determinacy

Turing Determinacy is trivial

Begin by noting that �-TD for any � is extremely weak both reversemathematically and in terms of consistency strength.

PropositionFor any class �, �-TD is true in REC, the !-model of RCA0 whose setsare just the recursive sets.

So �-TD cannot prove any theorem that implies the existence of anonrecursive set. These are essentially all the theorems studied in reversemathematics not provable in RCA0 (perhaps with some extra inductionaxioms).

Proposition

For any class �, �-TD is �11 conservative over RCA0.

So �-TD cannot prove the consistency even of RCA0 for any class �.Nonetheless there are interesting things to say.


Turing Determinacy





Proposition




Turing Determinacy





Proposition




Turing Determinacy





Proposition


So �-TD cannot prove the consistency even of RCA0 for any class �.

Nonetheless there are interesting things to say.


Turing Determinacy





Proposition


So �-TD cannot prove the consistency even of RCA0 for any class �.Nonetheless there are interesting things to say.Richard A. Shore (Cornell University ) Turing Determinacy 8/5/2015 21 / 33

Turing Determinacy

Turing Determinacy in RCA0 and ACA0

Proposition

RCA0` �02-TD.

Every nonempty �su¢ ciently closed��Z2 set contains a set of every Turingdegree above that of Z . Degree invariant sets are su¢ ciently closed.

Proposition

ACA0 ` �03-TD.

Also use Kuratowski�s representation of � sets by the trans�nite di¤erencehierarchy on � or � sets (MedSalam and Tanaka in ACA0). However,

Proposition

RCA0 0 �03-TD.

Consider a uniformly �02 initial segment of the Turing degrees of type !.The ones at the odd and even levels are each �03 de�nable.


Turing Determinacy


Proposition

RCA0` �02-TD.


Proposition

ACA0 ` �03-TD.


Proposition

RCA0 0 �03-TD.

Consider a uniformly �02 initial segment of the Turing degrees of type !.The ones at the odd and even levels are each �03 de�nable.


Turing Determinacy


Proposition

RCA0` �02-TD.


Proposition

ACA0 ` �03-TD.


Proposition

RCA0 0 �03-TD.

Consider a uniformly �02 initial segment of the Turing degrees of type !.The ones at the odd and even levels are each �03 de�nable.Richard A. Shore (Cornell University ) Turing Determinacy 8/5/2015 22 / 33

Turing Determinacy

Proving �03-TD and �04-TD

Theorem (essentially Harrington and Kechris [1975])

ATR0 ` �03-TD.

�04-TD needs Kuratowski representation (ACA0), an analysis of thecomplexity of containing a cone (on its face �12) and some extra induction.

Proposition (ATR0)

The predicate that (the formula de�ning) a �03 degree invariant set of realscontains a cone of degrees is �11.

TheoremATR0 + �11-TI0 ` �04-TD.

�11-TI0 is equivalent to �11-DC0 over ACA0 and so even with ATR0 weaker

than �11-CA0 as is then �04-TD.


Turing Determinacy



ATR0 ` �03-TD.


Proposition (ATR0)






Turing Determinacy



ATR0 ` �03-TD.


Proposition (ATR0)






Turing Determinacy



ATR0 ` �03-TD.


Proposition (ATR0)






Turing Determinacy

ATR0 0 �04-TD

TheoremATR0 0 �04-TD.

Essential ingredient for model of ATR0 + :�04-TD is !-incompleteness:

Theorem (H. Friedman)Let S be a recursive set of sentences of second order arithmetic containingACA0. If there exists a countable coded !-model of S, then there exists acountable coded !-model of S [ f:9countable coded !-model of Sg.

Use to carefully construct a sequenceMn of !-models of ATR0. Applycompactness to get (a nonstandard) model N̂ with elements for each oftheMn. Then take the (relatively) !-submodel N of N̂ generated by the(images of the)Mn. Prove it is a model of ATR0 and that in it we cande�ne the degrees of the even and the odd elements of the generatingsequence as �04 sets.Richard A. Shore (Cornell University ) Turing Determinacy 8/5/2015 24 / 33

Turing Determinacy

A reversal for �03 � TD; proving �04 � TD & �05 � TD

TheoremACA0 + �03-TD ` ATR0.

To prove 0(�) exists use Posner-Robinson and pseudojump inversion toanalyze fX j9� < �(0� � X �T W X g where X <T W X and X 0 �T W X .

Lemma (essentially Harrington and Kechris)

RCA0 ` �0n-Determinacy ! �0n+1-TD.

Lemma (essentially Martin)

�11-CA0 ` �0n-TD $ �0n+1-TD. (Also uses �13 conservativity of �

12-AC0

and �12-CA0 over �11-CA0.)

Corollary

�13-CA0 ` �04-TD & �05-TD.


Turing Determinacy

A reversal for �03 � TD; proving �04 � TD & �05 � TD

TheoremACA0 + �03-TD ` ATR0.

To prove 0(�) exists use Posner-Robinson and pseudojump inversion toanalyze fX j9� < �(0� � X �T W X g where X <T W X and X 0 �T W X .

Lemma (essentially Harrington and Kechris)

RCA0 ` �0n-Determinacy ! �0n+1-TD.

Lemma (essentially Martin)

�11-CA0 ` �0n-TD $ �0n+1-TD. (Also uses �13 conservativity of �

12-AC0

and �12-CA0 over �11-CA0.)

Corollary

�13-CA0 ` �04-TD & �05-TD.Richard A. Shore (Cornell University ) Turing Determinacy 8/5/2015 25 / 33

Turing Determinacy

Lower Bounds for �04-TD and �05-TD

Theorem�11-CA0 + �

04-TD ` �2 (the least �2 admissible ordinal) exists.

(Hardproof; extends MS[12] and MS[14].)

Corollary

�13-CA0 does not prove �04-TD. Indeed, �

11-CA0 + �

04-TD proves

�(�13-CA0) and hence much more than the consistency of �13-CA0.

Lemma (Martin)

�11-CA0 + �05-TD ` �0 (= ��(L� � Z2) exists.

Corollary (Martin)

Z2 does not prove �05-TD. Indeed, �11-CA0 + �

05-TD proves that for every

set Y there is a �(Z2) and hence much more than the consistency of Z2.


Turing Determinacy



04-TD ` �2 (the least �2 admissible ordinal) exists. (Hard

proof; extends MS[12] and MS[14].)

Corollary


11-CA0 + �

04-TD proves


Lemma (Martin)

�11-CA0 + �05-TD ` �0 (= ��(L� � Z2) exists.

Corollary (Martin)





Turing Determinacy





Corollary


11-CA0 + �

04-TD proves


Lemma (Martin)

�11-CA0 + �05-TD ` �0 (= ��(L� � Z2) exists.

Corollary (Martin)





Turing Determinacy





Corollary


11-CA0 + �

04-TD proves


Lemma (Martin)

�11-CA0 + �05-TD ` �0 (= ��(L� � Z2) exists.

Corollary (Martin)





Questions

Questions: Reverse Mathematical in Z2

There are several natural questions about Turing determinacy in Z2 leftopen here. We �rst point to two for which we expect that answers shouldrequire some new interesting models of fragments of Z2.

QuestionDoes WKL0 or some other known principle strictly between RCA0 andACA0 prove �03-TD?

QuestionDoes �03-TD (or �-TD for some larger �) prove ACA0 over WKL0?


Questions

A proof theoretically oriented variation asks about conservativity.

QuestionBoth WKL0 and �-TD are �11 conservative over RCA0. Is there a �(perhaps �03 ) such that WKL0 + �-TD is not �

11 conservative over RCA0?

Moving up the hierarchy we ask the following:

QuestionClarify the status of �04-TD over ACA0. In particular does ATR0 + �

11-TI0

(or equivalently �11-IND) or ATR0 with full induction prove �04-TD? If not,

does ACA0 + �04-TD prove �11-TI0?

QuestionDoes �04-TD (or �-TD for some larger �) prove �

11-CA0 over ATR0?


Questions






11-TI0




11-CA0 over ATR0?


Questions






11-TI0




11-CA0 over ATR0?


Questions






11-TI0




11-CA0 over ATR0?


Questions


Questions

Proof of Lower Bounds for �04-TD

Lemma

�11-CA0 + �04-TD ` �2 exists.

Consider the theory T = KP + \V = L"+8 (L is countable inside L +1) + no ordinal is �2-admissible:

Recall that �2 (the least �2 admissible ordinal) is the least � such thatL� \ R ��13-CA0. So this Lemma shows that �11-CA0 + �04-TD proves�(�13-CA0). For the sake of a contradiction assume �2 does not exist.

Now prove (essentially Martin) that

A = f� jL� � T and every member of L� is de�nable in L�g

is unbounded in the ordinals.


Questions

Proof of Lower Bounds for �04-TD

Lemma

�11-CA0 + �04-TD ` �2 exists.

Consider the theory T = KP + \V = L"+8 (L is countable inside L +1) + no ordinal is �2-admissible:

Recall that �2 (the least �2 admissible ordinal) is the least � such thatL� \ R ��13-CA0. So this Lemma shows that �11-CA0 + �04-TD proves�(�13-CA0). For the sake of a contradiction assume �2 does not exist.

Now prove (essentially Martin) that

A = f� jL� � T and every member of L� is de�nable in L�g

is unbounded in the ordinals.


Questions

Towards a contradiction, de�ne a �04 subset of 2N:

P = fX j 9T̂ [T̂ �T W X & T̂ is a complete extension of Twhose term modelMI is an !-model &8 ~T ( ~T �T X & ~T is a complete extension of Twhose term modelMII is an !-model !conditions RI new or RI 3 hold)]g.

W X is the pseudojump operator producing a low nonrecursive r.e. set.

Properties of the term models: Since every element ofMI andMII isde�nable by a real (because T says that every set is countable), we cancompare their elements by looking at the reals coding them. Thus, whenwe sayMI �MII, we mean that every element ofMI is coded by a realinMI which also belongs toMII. (As both are !-models, we cancon�dently talk about reals, i.e. subsets of !, being in one or both ofthem.)


Questions

Towards a contradiction, de�ne a �04 subset of 2N:

P = fX j 9T̂ [T̂ �T W X & T̂ is a complete extension of Twhose term modelMI is an !-model &8 ~T ( ~T �T X & ~T is a complete extension of Twhose term modelMII is an !-model !conditions RI new or RI 3 hold)]g.

W X is the pseudojump operator producing a low nonrecursive r.e. set.

Properties of the term models: Since every element ofMI andMII isde�nable by a real (because T says that every set is countable), we cancompare their elements by looking at the reals coding them. Thus, whenwe sayMI �MII, we mean that every element ofMI is coded by a realinMI which also belongs toMII. (As both are !-models, we cancon�dently talk about reals, i.e. subsets of !, being in one or both ofthem.)


Questions

Required properties of these conditions:

1 If one ofMI andMII is well-founded, then RI new holds if and onlyifMI is isomorphic to the well-founded part ofMII.

2 IfMI andMII are incomparable, then either RI 3 or RII 3 holds.3 If RI 3 holds, thenMII is ill-founded, and if If RII 3 holds thenMI isill-founded.

4 The conditions RI new and RI 3 are �03.

How this is accomplished will have to remain mysterious.


Questions

Required properties of these conditions:

1 If one ofMI andMII is well-founded, then RI new holds if and onlyifMI is isomorphic to the well-founded part ofMII.

2 IfMI andMII are incomparable, then either RI 3 or RII 3 holds.3 If RI 3 holds, thenMII is ill-founded, and if If RII 3 holds thenMI isill-founded.

4 The conditions RI new and RI 3 are �03.

How this is accomplished will have to remain mysterious.


Questions

Calculate that P is �04 as then so is its closure P̂ under �T . By �04-TD, P̂contains or is disjoint, from a cone. By Shoen�eld�s absoluteness theorem,the base z of the cone can be taken to be in L.

Let � be an admissible ordinal such that L� j= T and every element of L�is de�nable in L� and such that Z 2 L�. (Such an ordinal exists by theunboundedness result above which assumes that �2 does not exist.) LetTh� be the theory of L�. So, in particular Z ;Z 0 �T Tha.

Now argue �rst that Th� 2 P̂. Then argue that there is an X �T Z in P:As Z 0 �T Th� we can apply pseudojump inversion to get an X >T Z suchthat W X �T Th�. Argue that X 2 P . Now both Th� and X are Turingabove Z which is then not the base of a cone for P̂.


Questions





Questions





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