On hyper-arithmetic reflection principles

Turing-Taylor expansions for arithmetic theoriesThe Turing jump interpretation of transfinite provability logic

Hyper-arithmetic reflection

On hyper-arithmetic reflection principles

Joost J. Joosten

Universitat de Barcelona

Wednesday 30-09-2014Second International Wormshop, Mexico City

Joost J. Joosten On hyper-arithmetic reflection principles



In Memoriam: Grisha Mints




Turing progressionsThe logics GLPΛ and Ignatiev’s modelIgnatiev’s model and Turing-Taylor expansions

I Let T be some r.e. sound theory

I By Godel 2 we know that Con(T ) is independent of T

I So, we can add it and obtain an new sound theory

I We define the Turing(-Feferman) progression along a recursiveΓ of T as follows:

I

T0 := T ;Tα+1 := Tα + Con(Tα);Tλ :=

⋃α<λ Tα for limit λ < Γ.









I

T0 := T ;Tα+1 := Tα + Con(Tα);Tλ :=










I

T0 := T ;Tα+1 := Tα + Con(Tα);Tλ :=










I

T0 := T ;Tα+1 := Tα + Con(Tα);Tλ :=










I

T0 := T ;Tα+1 := Tα + Con(Tα);Tλ :=










I

T0 := T ;

Tα+1 := Tα + Con(Tα);Tλ :=










I

T0 := T ;Tα+1 := Tα + Con(Tα);

Tλ :=⋃α<λ Tα for limit λ < Γ.









I

T0 := T ;Tα+1 := Tα + Con(Tα);Tλ :=






I The obvious way of proving things about Turing progressionsis by transfinite induction.

I How can weak theories still prove interesting statementsabout Turing progressions?

I Schmerl (1978): reflexive transfinite induction



















I Transfinite induction: ∀α (∀β<α φ(β)→ φ(α)) → ∀α φ(α);

I Theorem EA proves reflexive transfinite induction (Schmerl)

If EA ` ∀α(2EA ∀β<α φ(β) → φ(α)

), then

EA ` ∀α φ(α).

I Proof By Lob’s rule

I Clearly, if

T ` ∀α(2T ∀β<α φ(β) → φ(α)

),

then alsoT ` 2T ∀α φ(α) → ∀α φ(α),








), then

EA ` ∀α φ(α).


I Clearly, if

T ` ∀α(2T ∀β<α φ(β) → φ(α)

),









), then

EA ` ∀α φ(α).


I Clearly, if

T ` ∀α(2T ∀β<α φ(β) → φ(α)

),









), then

EA ` ∀α φ(α).


I Clearly, if

T ` ∀α(2T ∀β<α φ(β) → φ(α)

),









), then

EA ` ∀α φ(α).


I Clearly, if

T ` ∀α(2T ∀β<α φ(β) → φ(α)

),

then also

T ` 2T ∀α φ(α) → ∀α φ(α),








), then

EA ` ∀α φ(α).


I Clearly, if

T ` ∀α(2T ∀β<α φ(β) → φ(α)

),






I We can generalize Turing progressions to stronger notions ofconsistency.

I For n ∈ ω:

I We will denote “provable in T using all true Πn sentences” by[n]T

(of logical complexity

Σ0n+1

)

I The dual notion “consistent with T and all true Πn

sentences” is denoted 〈n〉T .

(

Π0n+1

)

I Then

I T i0 := T ;

I T iα+1 := T i

α ∪ {〈i〉T iα>};

I Tλ :=⋃α<λ Tαfor limit λ.






I For n ∈ ω:



Σ0n+1

)



(

Π0n+1

)

I Then

I T i0 := T ;

I T iα+1 := T i

α ∪ {〈i〉T iα>};







I For n ∈ ω:



Σ0n+1

)



(

Π0n+1

)

I Then

I T i0 := T ;

I T iα+1 := T i

α ∪ {〈i〉T iα>};







I For n ∈ ω:

I We will denote “provable in T using all true Πn sentences” by[n]T (of logical complexity

Σ0n+1

)



(

Π0n+1

)

I Then

I T i0 := T ;

I T iα+1 := T i

α ∪ {〈i〉T iα>};







I For n ∈ ω:

I We will denote “provable in T using all true Πn sentences” by[n]T (of logical complexity Σ0

n+1)



(

Π0n+1

)

I Then

I T i0 := T ;

I T iα+1 := T i

α ∪ {〈i〉T iα>};







I For n ∈ ω:


n+1)



(

Π0n+1

)I Then

I T i0 := T ;

I T iα+1 := T i

α ∪ {〈i〉T iα>};







I For n ∈ ω:


n+1)


sentences” is denoted 〈n〉T . (

Π0n+1

)

I Then

I T i0 := T ;

I T iα+1 := T i

α ∪ {〈i〉T iα>};







I For n ∈ ω:


n+1)


sentences” is denoted 〈n〉T . ( Π0n+1 )

I Then

I T i0 := T ;

I T iα+1 := T i

α ∪ {〈i〉T iα>};







I For n ∈ ω:


n+1)



I Then

I T i0 := T ;

I T iα+1 := T i

α ∪ {〈i〉T iα>};







I For n ∈ ω:


n+1)



I ThenI T i

0 := T ;

I T iα+1 := T i

α ∪ {〈i〉T iα>};







I For n ∈ ω:


n+1)



I ThenI T i

0 := T ;I T i

α+1 := T iα ∪ {〈i〉T i

α>};







I For n ∈ ω:


n+1)



I ThenI T i

0 := T ;I T i

α+1 := T iα ∪ {〈i〉T i

α>};






I Poly-modal provability logics turn out to be suitably wellequipped to talk about Turing progressions

I Already just the language with one modality [0] is expressive

I Godel II: 3T> → ¬2T3T>I Godel II: 2T (2T⊥ → ⊥)→ 2T⊥I For n ∈ N we see Tn ≡ T + 3n

T>I Transfinite progressions are not expressible in the modal

language with just one modal operator.

I However:

I Proposition: T + 〈1〉T> is a Π1 conservative extension ofT + {〈0〉kT> | k ∈ ω}.










I However:








I Godel II: 3T> → ¬2T3T>

I Godel II: 2T (2T⊥ → ⊥)→ 2T⊥I For n ∈ N we see Tn ≡ T + 3n



I However:








I Godel II: 3T> → ¬2T3T>I Godel II: 2T (2T⊥ → ⊥)→ 2T⊥

I For n ∈ N we see Tn ≡ T + 3nT>

I Transfinite progressions are not expressible in the modallanguage with just one modal operator.

I However:









T>

I Transfinite progressions are not expressible in the modallanguage with just one modal operator.

I However:











I However:











I However:











I However:






DefinitionThe logic GLPΛ is the propositional normal modal logic that hasfor each ξ < Λ a modality [ξ] and is axiomatized by the followingschemata:

[ξ](A→ B)→ ([ξ]A→ [ξ]B)[ξ]([ξ]A→ A)→ [ξ]A〈ξ〉A→ [ζ]〈ξ〉A for ξ < ζ,[ξ]A→ [ζ]A for ξ < ζ.

The rules of inference are Modus Ponens and necessitation foreach modality: ψ

[ζ]ψ .








[ζ]ψ .








[ζ]ψ .





I GLP0Λ denotes the closed fragment (no propositional variables)

I Iterated consistency statements in GLP0Λ are called worms

I 〈ξ0〉 . . . 〈ξn〉>I We write W for the class of all worms

I We write Wξ for the class of all worms all of whose modalitiesare at least ξ

I We can define natural orderings <ξ on W by

A <ξ B :⇔ GLP ` B → 〈ξ〉A










A <ξ B :⇔ GLP ` B → 〈ξ〉A







I 〈ξ0〉 . . . 〈ξn〉>

I We write W for the class of all worms



A <ξ B :⇔ GLP ` B → 〈ξ〉A










A <ξ B :⇔ GLP ` B → 〈ξ〉A










A <ξ B :⇔ GLP ` B → 〈ξ〉A










A <ξ B :⇔ GLP ` B → 〈ξ〉A





A <ξ B :⇔ GLP ` B → 〈ξ〉A

I For <0 defines a well-order on the class of worms moduloprovable GLP equivalence.

(Beklemishev, Fernandez Duque, JjJ)

I For <ξ with ξ > 0 the relation is no longer linear (mod prov.equivalence) but is still well-founded

(infinite anti-chains)

I Definition By oα(A) we denote the order type of A under <α

and we write o(A) instead of o0(A).

I Worms of GLPω are known to be useful for Turingprogressions:

I Proposition (Beklemishev) For each ordinal α < ε0 there issome GLPω-worm A such that o(A) = α, and T + A is Π1

equivalent to Tα.





A <ξ B :⇔ GLP ` B → 〈ξ〉A

I For <0 defines a well-order on the class of worms moduloprovable GLP equivalence.(Beklemishev, Fernandez Duque, JjJ)







equivalent to Tα.





A <ξ B :⇔ GLP ` B → 〈ξ〉A








equivalent to Tα.





A <ξ B :⇔ GLP ` B → 〈ξ〉A


I For <ξ with ξ > 0 the relation is no longer linear (mod prov.equivalence) but is still well-founded (infinite anti-chains)





equivalent to Tα.





A <ξ B :⇔ GLP ` B → 〈ξ〉A







equivalent to Tα.





A <ξ B :⇔ GLP ` B → 〈ξ〉A



I Definition By oα(A) we denote the order type of A under <αand we write o(A) instead of o0(A).



equivalent to Tα.





A <ξ B :⇔ GLP ` B → 〈ξ〉A






equivalent to Tα.





A <ξ B :⇔ GLP ` B → 〈ξ〉A






equivalent to Tα.





I For the first part of this talk we shall focus on GLPω

I For a worm A we define hn(A) as the n head as the largestpart on the left of A where all modalities are at least n

I Example: h2(34245) = 34245 and h3(34) = 34

I We define an Ignatiev sequence to be a sequence ~A = {Ai} ofworms so that

I Each An ∈ Bn

I An+1 ≤n+1 hn+1(An)

I Example: 〈22022, 22〉I but also 〈22022, 2〉





I For the first part of this talk we shall focus on GLPωI For a worm A we define hn(A) as the n head as the largest

part on the left of A where all modalities are at least n

I Example: h2(34245) = 34245 and h3(34) = 34


I Each An ∈ Bn









I Example: h2(34245) = 34245 and h3(34) = 34


I Each An ∈ Bn









I Example: h2(34245) = 34245 and h3(34) = 34


I Each An ∈ Bn









I Example: h2(34245) = 34245 and h3(34) = 34


I Each An ∈ Bn









I Example: h2(34245) = 34245 and h3(34) = 34


I Each An ∈ Bn









I Example: h2(34245) = 34245 and h3(34) = 34


I Each An ∈ Bn


I Example: 〈22022, 22〉

I but also 〈22022, 2〉







I Example: h2(34245) = 34245 and h3(34) = 34


I Each An ∈ Bn







I For two Ignatiev sequences ~A and ~B we define an accessibilityrelation <n:

I ~A <n~B if and only if

I Am = Bm for all m < nI An <n Bn

I Example: 〈202, 22〉 >1 〈202, 2〉I but also 〈202, 22〉 >1 〈202, 2, 2〉















I Am = Bm for all m < n

I An <n Bn

















I Example: 〈202, 22〉 >1 〈202, 2〉

I but also 〈202, 22〉 >1 〈202, 2, 2〉













I Let I denote the set of all Ignatiev sequences

I We define a Kripke frame:

〈I, {>n}n∈ω〉

I We shall denote this frame also by II We define by ~A >, for no ~A, ~A ⊥.

I commutes with Boolean connectives: ~A φ ∧ ψ if and onlyif ~A φ and ~A ψ, etc

I ~A 〈n〉φ if and only if there is some ~B with ~A >n~B so that

~B φ

I Theorem GLP0ω ` φ ⇔ I ` φ

I Proof by a p-morphic embedding of this structure into thegeneralization of Ignatiev’s model.

I Let’s see a picture







〈I, {>n}n∈ω〉




~B φ










〈I, {>n}n∈ω〉

I We shall denote this frame also by I

I We define by ~A >, for no ~A, ~A ⊥.



~B φ










〈I, {>n}n∈ω〉




~B φ










〈I, {>n}n∈ω〉




~B φ










〈I, {>n}n∈ω〉




~B φ










〈I, {>n}n∈ω〉




~B φ










〈I, {>n}n∈ω〉




~B φ










〈I, {>n}n∈ω〉




~B φ








I We define the Πn+1 proof-theoretic ordinal of a theory U asfollows:

I |U|Πn+1 = sup{ξ | T nξ ⊆ U}.

I For U a arithmetical theory we define its Turing-Taylorexpansion by

I tt(U) :=⋃∞

n=0 T n|U|Πn+1

I In case U ≡ tt(U) we say that U has a convergentTuring-Taylor expansion.








I tt(U) :=⋃∞

n=0 T n|U|Πn+1









I tt(U) :=⋃∞

n=0 T n|U|Πn+1









I tt(U) :=⋃∞

n=0 T n|U|Πn+1









I tt(U) :=⋃∞

n=0 T n|U|Πn+1






I We will now link Ignatiev’s model to Turing-Taylor expansions

I Let us recall:

I T i0 := T ;

I T iα+1 := T i

α ∪ {〈i〉T iα>};


I We shall use the ordinal notation system 〈Bn, <n〉 to label theTuring progression based on n-consistency

I Thus, T 13 denotes T 1

ωω ,

I and T 23 denotes T 2

ω


1






I Let us recall:

I T i0 := T ;

I T iα+1 := T i

α ∪ {〈i〉T iα>};




ωω ,


ω


1






I Let us recall:

I T i0 := T ;

I T iα+1 := T i

α ∪ {〈i〉T iα>};




ωω ,


ω


1






I Let us recall:

I T i0 := T ;

I T iα+1 := T i

α ∪ {〈i〉T iα>};




ωω ,


ω


1






I Let us recall:

I T i0 := T ;

I T iα+1 := T i

α ∪ {〈i〉T iα>};




ωω ,


ω


1






I Let us recall:

I T i0 := T ;

I T iα+1 := T i

α ∪ {〈i〉T iα>};




ωω ,


ω


1






I Let us recall:

I T i0 := T ;

I T iα+1 := T i

α ∪ {〈i〉T iα>};




ωω ,


ω


1






I Let us recall:

I T i0 := T ;

I T iα+1 := T i

α ∪ {〈i〉T iα>};




ωω ,


ω


1






I Let us recall:

I T i0 := T ;

I T iα+1 := T i

α ∪ {〈i〉T iα>};




ωω ,


ω


1





I TheoremFor each worm A : T + A ≡

⋃∞n=0 T n

hn(A)


⋃∞n=0 T n

A

I Compare this to

f (x) :=∞∑

n=0

f (n)(0)

n!xn






⋃∞n=0 T n

hn(A)


⋃∞n=0 T n

A

I Compare this to

f (x) :=∞∑

n=0

f (n)(0)

n!xn






⋃∞n=0 T n

hn(A)


⋃∞n=0 T n

A

I Compare this to

f (x) :=∞∑

n=0

f (n)(0)

n!xn





I Theorem The Ignatiev sequences exactly correspond to thosesub-theories of PA that have a convergent Turing-Taylorexpansion

I That is, for each such theory U, we have that tt(U) ∈ II and for each ~A ∈ I, there is a theory U so that tt(U) = ~A

I This yields a roadmap to conservation results!






I That is, for each such theory U, we have that tt(U) ∈ I

I and for each ~A ∈ I, there is a theory U so that tt(U) = ~A




















I We proof of the theorem uses three main results

I We shall see why worms are better than the more familiarordinal notations in this context

I For each number n and each GLPω worm A,GLPω ` A↔ hn(A) ∧ rn(A)

(here, rn(A) denotes the nremainder of A so that A = hn(A)rn(A))

I For each worm A ∈Wn we have T + A ≡n T nA (Beklemishev)

I For each worm A : T + A ≡⋃∞

n=0 T nhn(A) (JjJ)

I Corollaries:

I For each worm A ∈Wn we have T + nA ≡ T nnA

I For each worm A ∈Wn T nA ` T m

A for m < n











n=0 T nhn(A) (JjJ)

I Corollaries:



A for m < n











n=0 T nhn(A) (JjJ)

I Corollaries:



A for m < n







I For each number n and each GLPω worm A,GLPω ` A↔ hn(A) ∧ rn(A) (here, rn(A) denotes the nremainder of A so that A = hn(A)rn(A))



n=0 T nhn(A) (JjJ)

I Corollaries:



A for m < n










n=0 T nhn(A) (JjJ)

I Corollaries:



A for m < n










n=0 T nhn(A) (JjJ)

I Corollaries:



A for m < n










n=0 T nhn(A) (JjJ)

I Corollaries:



A for m < n










n=0 T nhn(A) (JjJ)

I Corollaries:I For each worm A ∈Wn we have T + nA ≡ T n

nA


A for m < n










n=0 T nhn(A) (JjJ)

I Corollaries:I For each worm A ∈Wn we have T + nA ≡ T n

nAI For each worm A ∈Wn T n

A ` T mA for m < n





I

tt(U) denotes 〈|U|Π01, |U|Π0

2, |U|Π0

3, . . .〉.

I Likewise, with every sequence ~α = 〈α0, α1, . . .〉 of ordinalsbelow ε0 we can naturally associate a sub theory (~α)tt of PAas follows

(~α)tt :=∞⋃

n=0

EAnαn.

I Likewise, with every sequence ~A = 〈A0,A1, . . .〉 of GLPωworms we can naturally associate a sub theory (~A)tt of PA asfollows

(~α)tt :=∞⋃

n=0

EAnAn.





I


2, |U|Π0

3, . . .〉.


(~α)tt :=∞⋃

n=0

EAnαn.


(~α)tt :=∞⋃

n=0

EAnAn.





I


2, |U|Π0

3, . . .〉.


(~α)tt :=∞⋃

n=0

EAnαn.


(~α)tt :=∞⋃

n=0

EAnAn.





I The monomials in Turing-Taylor progressions are the T nA

I They are not entirely independent!

I Example: T 11 + T 0

01 ≡ T 11 + T 0

101.

I T 11 ≡ T + 〈1〉>, and

I T 001 ≡ T 0

01.

I Thus, T 11 + T 0

01 ≡ T + 〈1〉>+ 〈0〉〈1〉>.

I Equivalent to T + 〈1〉〈0〉〈1〉>I Which is in turn equivalent to T 1

1 + T 0101

I In the classical notation system this reads

T 11 + T 0

ω+1 ≡ T 11 + T 0

ω·2

I The example shows that in general tt((~A)tt) 6= ~A !

tt((01, 1)tt) = (101, 1) 6= (01, 1)








01 ≡ T 11 + T 0

101.

I T 11 ≡ T + 〈1〉>, and

I T 001 ≡ T 0

01.

I Thus, T 11 + T 0

01 ≡ T + 〈1〉>+ 〈0〉〈1〉>.


1 + T 0101


T 11 + T 0

ω+1 ≡ T 11 + T 0

ω·2


tt((01, 1)tt) = (101, 1) 6= (01, 1)








01 ≡ T 11 + T 0

101.

I T 11 ≡ T + 〈1〉>, and

I T 001 ≡ T 0

01.

I Thus, T 11 + T 0

01 ≡ T + 〈1〉>+ 〈0〉〈1〉>.


1 + T 0101


T 11 + T 0

ω+1 ≡ T 11 + T 0

ω·2


tt((01, 1)tt) = (101, 1) 6= (01, 1)








01 ≡ T 11 + T 0

101.

I T 11 ≡ T + 〈1〉>, and

I T 001 ≡ T 0

01.

I Thus, T 11 + T 0

01 ≡ T + 〈1〉>+ 〈0〉〈1〉>.


1 + T 0101


T 11 + T 0

ω+1 ≡ T 11 + T 0

ω·2


tt((01, 1)tt) = (101, 1) 6= (01, 1)








01 ≡ T 11 + T 0

101.

I T 11 ≡ T + 〈1〉>, and

I T 001 ≡ T 0

01.

I Thus, T 11 + T 0

01 ≡ T + 〈1〉>+ 〈0〉〈1〉>.


1 + T 0101


T 11 + T 0

ω+1 ≡ T 11 + T 0

ω·2


tt((01, 1)tt) = (101, 1) 6= (01, 1)








01 ≡ T 11 + T 0

101.

I T 11 ≡ T + 〈1〉>, and

I T 001 ≡ T 0

01.

I Thus, T 11 + T 0

01 ≡ T + 〈1〉>+ 〈0〉〈1〉>.


1 + T 0101


T 11 + T 0

ω+1 ≡ T 11 + T 0

ω·2


tt((01, 1)tt) = (101, 1) 6= (01, 1)








01 ≡ T 11 + T 0

101.

I T 11 ≡ T + 〈1〉>, and

I T 001 ≡ T 0

01.

I Thus, T 11 + T 0

01 ≡ T + 〈1〉>+ 〈0〉〈1〉>.

I Equivalent to T + 〈1〉〈0〉〈1〉>

I Which is in turn equivalent to T 11 + T 0

101


T 11 + T 0

ω+1 ≡ T 11 + T 0

ω·2


tt((01, 1)tt) = (101, 1) 6= (01, 1)








01 ≡ T 11 + T 0

101.

I T 11 ≡ T + 〈1〉>, and

I T 001 ≡ T 0

01.

I Thus, T 11 + T 0

01 ≡ T + 〈1〉>+ 〈0〉〈1〉>.


1 + T 0101


T 11 + T 0

ω+1 ≡ T 11 + T 0

ω·2


tt((01, 1)tt) = (101, 1) 6= (01, 1)








01 ≡ T 11 + T 0

101.

I T 11 ≡ T + 〈1〉>, and

I T 001 ≡ T 0

01.

I Thus, T 11 + T 0

01 ≡ T + 〈1〉>+ 〈0〉〈1〉>.


1 + T 0101


T 11 + T 0

ω+1 ≡ T 11 + T 0

ω·2


tt((01, 1)tt) = (101, 1) 6= (01, 1)








01 ≡ T 11 + T 0

101.

I T 11 ≡ T + 〈1〉>, and

I T 001 ≡ T 0

01.

I Thus, T 11 + T 0

01 ≡ T + 〈1〉>+ 〈0〉〈1〉>.


1 + T 0101


T 11 + T 0

ω+1 ≡ T 11 + T 0

ω·2


tt((01, 1)tt) = (101, 1) 6= (01, 1)








01 ≡ T 11 + T 0

101.

I T 11 ≡ T + 〈1〉>, and

I T 001 ≡ T 0

01.

I Thus, T 11 + T 0

01 ≡ T + 〈1〉>+ 〈0〉〈1〉>.


1 + T 0101


T 11 + T 0

ω+1 ≡ T 11 + T 0

ω·2

I The example shows that in general tt((~A)tt) 6= ~A !tt((01, 1)tt) = (101, 1) 6= (01, 1)






01 ≡ T 11 + T 0

101.

I With a slightly more involved reasoning, we can prove

I Lemma 1: Let A ∈ Sn+1 and B ∈ Sn. We have that

I

T n+1A + T n

nB ≡n+1 T + AnB,

and

I

T n+1A + T n

nB ≡n T nAnB .

I This nicely illustrates that worms are often better than Cantor

I Using these ingredients one easily proves

I Theorem If U is some sub-theory of PA with a convergentTuring-Taylor expansion, so that U 6≡0 PA, then tt(U) definesa point in I.






01 ≡ T 11 + T 0

101.



I

T n+1A + T n

nB ≡n+1 T + AnB,

and

I

T n+1A + T n

nB ≡n T nAnB .









01 ≡ T 11 + T 0

101.



I

T n+1A + T n

nB ≡n+1 T + AnB,

and

I

T n+1A + T n

nB ≡n T nAnB .









01 ≡ T 11 + T 0

101.



I

T n+1A + T n

nB ≡n+1 T + AnB,

and

I

T n+1A + T n

nB ≡n T nAnB .









01 ≡ T 11 + T 0

101.



I

T n+1A + T n

nB ≡n+1 T + AnB,

and

I

T n+1A + T n

nB ≡n T nAnB .









01 ≡ T 11 + T 0

101.



I

T n+1A + T n

nB ≡n+1 T + AnB,

and

I

T n+1A + T n

nB ≡n T nAnB .









01 ≡ T 11 + T 0

101.



I

T n+1A + T n

nB ≡n+1 T + AnB,

and

I

T n+1A + T n

nB ≡n T nAnB .









01 ≡ T 11 + T 0

101.



I

T n+1A + T n

nB ≡n+1 T + AnB,

and

I

T n+1A + T n

nB ≡n T nAnB .







The FGH Theorem and generalizationsThe Turing-jump interpretation of transfinite provability logics

I We would like to extend the results of the first section beyondfirst order

I A central ingredient: syntactical complexity classes

I Like in the truth interpretation of GLP

I Omega-rule interpretation is slightly better

I However, does not tie up with the Turing jump hierarchy

I Friedman, Godlfarb and Harrington come to the rescue!























































I Definition (Witness-comparison relation)

For φ := ∃x φ0(x) and ψ := ∃x ψ0(x) we define

φ ≤ ψ := ∃x (φ0(x) ∧ ∀ y<x ¬ψ0(x)) and,φ < ψ := ∃x (φ0(x) ∧ ∀ y≤x ¬ψ0(x)).

I Theorem (Rosser’s Theorem)

Let T be a consistent c.e. theory extending EA. There is someρ ∈ Σ0

1 which is undecidable in T . That is,

T 0 ρ and,T 0 ¬ρ.

I Proof Consider ρ↔ ¬(2ρ < 2¬ρ).I I find it utterly amazing that something sensible can be

proven using the witness comparison techniques!



















































I Proof Consider ρ↔ ¬(2ρ < 2¬ρ).

I I find it utterly amazing that something sensible can beproven using the witness comparison techniques!













proven using the witness comparison techniques!Joost J. Joosten On hyper-arithmetic reflection principles




LemmaLet A and B be some formulas of logical complexity Σ0

n+1 forn < ω.

1. Both A < B and A ≤ B are of complexity Σ0n+1;

2. EA ` A ∧ ¬B → (A < B) ;

3. EA ` (A < B)→ (A ≤ B);

4. EA ` (A ≤ B)→ A;

5. EA ` (A ≤ B)→ ¬(B < A) and consequently;

6. EA ` (A < B)→ ¬(B ≤ A);

7. EA ` [(B ≤ B) ∨ (A ≤ A)] → [(A ≤ B) ∨ (B < A)];

8. EA ` A ∧ ¬(A ≤ B)→ B.





I Theorem (FGH theorem)

Let T be any computably enumerable theory extending EA. Foreach σ ∈ Σ0

1 we have that there is some ρ ∈ Σ01 so that

EA ` 3T> →(σ ↔ 2Tρ

).

I Proof.Consider the fixpoint ρ for which EA ` ρ ↔ (σ ≤ 2ρ).

I This shows us that we can express a syntactical class usingprovability logics!

I We wish to stretch this further








EA ` 3T> →(σ ↔ 2Tρ

).











EA ` 3T> →(σ ↔ 2Tρ

).











EA ` 3T> →(σ ↔ 2Tρ

).











EA ` 3T> →(σ ↔ 2Tρ

).











EA ` 3T> →(σ ↔ 2Tρ

).








I Visser’s proof used A→ A ≤ A for A ∈ Σ1.

LemmaLet A ∈ Σ0

n+1, then the schema A→ (A ≤ A) is over EA provablyequivalent to the least-number principle for ∆0

n formulas.

I This can be avoided

I so FGH is generalizable over a weak base theory.






LemmaLet A ∈ Σ0


n formulas.








LemmaLet A ∈ Σ0


n formulas.








LemmaLet A ∈ Σ0


n formulas.







I By [n]TrueT we will denote the formalization of the predicate

“provable in T together with all true Π0n sentences”.

I It is well-known that for recursive theories T we can write[n]True

T by a Σ0n+1-formula.

I Also, we have provable Σ0n completeness for these predicates,

that is:

I propositionLet T be a computable theory extending EA and let φ be aΣ0

n+1 formula. We have that

EA ` φ→ [n]TrueT φ.










that is:













that is:













that is:








TheoremLet T be any computably enumerable theory extending EA and letn < ω. For each σ ∈ Σ0

n+1 we have that there is some ρn ∈ Σ0n+1

so thatEA ` 〈n〉True

T > →(σ ↔ [n]True

T ρn

).

I proof The proof runs entirely analogue to the proof of theclassical FGH theorem.

I Thus, for each number n we consider the fixpoint ρn so thatEA ` ρn ↔ (σ ≤ [n]True

T ρn).I Just using Σ0

n+1-completeness now

I Corollary

Let T be a c.e. theory extending EA and let n ∈ N. For eachformulas ϕ,ψ there is some σ ∈ Σ0

n+1 so that

T ` ([n]TrueT ϕ ∨ [n]True

T ψ) ↔ [n]TrueT σ.









T ρn

).





I Corollary


n+1 so that











T ρn

).



T ρn).

I Just using Σ0n+1-completeness now

I Corollary


n+1 so that











T ρn

).





I Corollary


n+1 so that











T ρn

).





I Corollary


n+1 so that











T ρn

).





I Corollary


n+1 so that







I The [n]True predicates tie up with the arithmetical hierarchy:

I LemmaLet T be any c.e. theory and let A ⊆ N. The following areequivalent

1. A is c.e. in ∅(n);

2. A is 1-1 reducible to ∅(n+1);

3. A is definable on the standard model by a Σ0n+1 formula;

4. A is definable on the standard model by a formula of the form[n]True

T ρ(x);


T ρ(x) where ρ(x) ∈ Σ0n+1;







1. A is c.e. in ∅(n);




T ρ(x);









1. A is c.e. in ∅(n);




T ρ(x);









1. A is c.e. in ∅(n);




T ρ(x);









1. A is c.e. in ∅(n);




T ρ(x);









1. A is c.e. in ∅(n);




T ρ(x);









1. A is c.e. in ∅(n);




T ρ(x);









1. A is c.e. in ∅(n);




T ρ(x);







I [n +1]OmegaT ϕ := ∃ψ

(∀x [n]Omega

T ψ(x) ∧ 2T (∀x ψ(x)→ ϕ))

I [n]OmegaT is a Σ0

2n+1-formula.

I LemmaLet T be a computable theory extending EA and let φ be a Σ0

2n+1

formula. We have that

EA ` φ→ [n]OmegaT φ.

Proof.By an external induction on n where each inductive step requiresthe application of an additional omega-rule.





Corollary

Let T be any computably enumerable theory extending EA and letn < ω. For each σ ∈ Σ0

2n+1 we have that there is some ρn ∈ Σ02n+1

so thatEA ` 〈n〉Omega

T > →(σ ↔ [n]Omega

T ρn

).





I LemmaLet T be any c.e. theory, let n be a natural number, and letA ⊆ N. The following are equivalent


2. A is definable on the standard model by a formula of the form[n]Omega

T ρ(x);

I Runs out of phase!

I We wish to use the best of both worlds








T ρ(x);










T ρ(x);










T ρ(x);










T ρ(x);










T ρ(x);







I

[0]2Tφ := 2Tφ, and

[n + 1]2Tφ := 2Tφ ∨ ∃ψ∨

0≤m≤n

(〈m〉2Tψ ∧ 2(〈m〉2Tψ → φ)

).





proposition

Let T be a sound c.e. theory extending EA. We have for all n ∈ Nthat

1. EA ` ∀ϕ([n]2Tϕ → [n]True

T ϕ);

2. EA ` 〈n〉TrueT > → ∀ϕ

([n + 1]2Tϕ↔ [n + 1]True

T ϕ);

3. EA ` [n]TrueT

(∀ϕ([n]2Tϕ↔ [n]True

T ϕ) )

;

4. N |= ∀ϕ([n]2Tϕ↔ [n]True

T ϕ).





TheoremLet T be a c.e. theory. We have for all A ⊆ N that the followingare equivalent

1. A is c.e. in ∅(n);


3. A is definable on the standard model by a formula of the form[n]2Tρ(x);





I We now generalize to the transfinte

I fixing a well-behaved ordinal

I TheoremThe logic GLPΛ is sound for strong enough theories T under theinterpretation 2 7→ [λ]2,ΛT .

























DefinitionLet T be a c.e. theory. We define

- ∆20 := Σ2

0 := Π20 := ∆0

0;

- Σ2α+1 = Σ2

α ∪ Π2α ∪ {[α]2Tϕ(x) | ϕ(x) ∈ Form} for α > 0;

- Π2α+1 = Σ2

α ∪ Π2α ∪ {〈α〉2Tϕ(x) | ϕ(x) ∈ Form} for α > 0;

- Σ2λ := Π2

λ :=⋃α<λ

Σ2α for λ ∈ Lim.




I Theorem/conjecture Let T be any c.e. theory, let ξ < Λ fora natural ordinal notation system, and let A ⊆ N. Thefollowing are equivalent

1. A is c.e. in ∅(ξ);

2. A is 1-1 reducible to ∅(ξ+1);

3. A is definable on the standard model by a formula of the form[ξ]2Tρ(x);

I No longer runs out of phase

I Theorem/conjecture: So all the stuff about Turingprogressions can be generalized in a straight-forward fashion.





1. A is c.e. in ∅(ξ);









1. A is c.e. in ∅(ξ);









1. A is c.e. in ∅(ξ);









1. A is c.e. in ∅(ξ);









1. A is c.e. in ∅(ξ);








DefinitionLet Γ be a class of formulas. For ordinals α, β < Λ and T ac.e. theory we define β−RFNΛ

T (Γ) to be the schema [β]2Tϕ→ ϕ forϕ ∈ Γ.

Instead of writing 0−RFNΛT (Γ) we shall just write RFNΛ

T (Γ).

We can now easily state and prove various equivalences betweenconsistency statements and reflection principles.




Theorem

Let T be a c.e. theory containing ECA0.

1. ECA0 ` RFNΛT (Π2

α+1) ≡ 〈α〉2T>;

2. For β ≤ α, we have ECA0 ` β−RFNΛT (Π2

α+1) ≡ 〈α〉2T>;

3. For β > α we have that ECA0 ` β−RFNΛT (Π2

α+1) ≡ 〈β〉2T>;

4. For β > α we have thatECA0 ` β−RFNΛ

T (Π2α+1) ≡ 〈max{α, β}〉2T>.


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On hyper-arithmetic reflection principles

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