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Incompleteness for higher order arithmetic: Aspecific example from set theory

Yong Cheng

School of PhilosophyWuhan University, Wuhan, China

Yong Cheng Incompleteness for higher order arithmetic: A specific example from set theory

Godel’s incompleteness theorem

Two goals of Hilbert’s program:

Completeness A proof that all true mathematical statements canbe proved in the formalism of mathematics.

Consistency A proof that no contradiction can be obtained in theformalism of mathematics using only ”finitistic”reasoning about finite mathematical objects.

Theorem (Godel-Rosser)

(1) Godel-Rosser first incompleteness theorem (G1): If T is arecursively axiomatized consistent extension of PA, then T isnot complete.

(2) Godel’s second incompleteness theorem (G2): If T is arecursively axiomatized consistent extension of PA, then theconsistency of T is not provable in T .


Godel’s incompleteness theorem

Two goals of Hilbert’s program:

Completeness A proof that all true mathematical statements canbe proved in the formalism of mathematics.

Consistency A proof that no contradiction can be obtained in theformalism of mathematics using only ”finitistic”reasoning about finite mathematical objects.

Theorem (Godel-Rosser)

(1) Godel-Rosser first incompleteness theorem (G1): If T is arecursively axiomatized consistent extension of PA, then T isnot complete.

(2) Godel’s second incompleteness theorem (G2): If T is arecursively axiomatized consistent extension of PA, then theconsistency of T is not provable in T .


Mathematical example of G1 for PA

Godel constructs a true sentence about arithmetic which isnot provable in PA.

From the viewpoint of classic mathematics, Godel’s sentenceis artificial and has no real mathematical content.

Question

Could we find a sentence about arithmetic with interestingmathematical contents which is independent of PA?

Theorem (Paris-Harrington)

If PA is consistent, then there exists a sentence φ of combinatorialcontents such that N |= φ, but φ is independent of PA whereN = (N,+, ·).





Question








Question





Definition of higher order arithmetic

Definition

(1) Z2 = ZFC−+ Every set is countable.a

(2) Z3 = ZFC− + P(ω) exists + Every set is of cardinality ≤ i1.

(3) Z4 = ZFC− + P(P(ω)) exists + Every set is of cardinality≤ i2.

aZFC− denotes ZFC with the Power Set Axiom deleted and Collectioninstead of Replacement.

Corollary

If Z2 is consistent, then there is a true sentence about analysiswhich is not provable in Z2.


Definition of higher order arithmetic

Definition

(1) Z2 = ZFC−+ Every set is countable.a

(2) Z3 = ZFC− + P(ω) exists + Every set is of cardinality ≤ i1.

(3) Z4 = ZFC− + P(P(ω)) exists + Every set is of cardinality≤ i2.

aZFC− denotes ZFC with the Power Set Axiom deleted and Collectioninstead of Replacement.

Corollary

If Z2 is consistent, then there is a true sentence about analysiswhich is not provable in Z2.


Relativized Hilbert’s program to Z2

Fact

Many classic mathematical theorems about analysis which areexpressible in Z2 are provable in Z2.

Question

Relativized Hilbert’s program to Z2 Is Z2 complete for classicmathematical theorems expressible in Z2?

Motivation In this talk, I give a counterexample from set theoryfor this question which is expressible in Z2 but notprovable in Z2.


Relativized Hilbert’s program to Z2

Fact

Many classic mathematical theorems about analysis which areexpressible in Z2 are provable in Z2.

Question

Relativized Hilbert’s program to Z2 Is Z2 complete for classicmathematical theorems expressible in Z2?

Motivation In this talk, I give a counterexample from set theoryfor this question which is expressible in Z2 but notprovable in Z2.


Martin-Harrington Theorem

All of the followings are expressible in Z2 and provable in ZF:

Lightface Martin’s theorem 0] exists implies Det(Σ11).

Lightface Harrington’s theorem Det(Σ11) implies 0] exists.

Lightface Martin-Harrington theorem Det(Σ11) if and only if 0]

exists.

Boldface Martin’s theorem If for any real x , x ] exists, thenDet(Σ1

1) holds.

Boldface Harrington’s theorem Det(Σ11) implies for any real x , x ]

exists.

Boldface Martin-Harrington theorem Det(Σ11) if and only if for

any real x , x ] exists.







exists.


1) holds.


exists.









exists.


1) holds.


exists.









exists.


1) holds.


exists.









exists.


1) holds.


exists.









exists.


1) holds.


exists.









exists.


1) holds.


exists.




Analysis of Lightface Harrington’s theorem

Definition

We let Harrington’s Principle, HP for short, denote the followingstatement: there is a real x such that if α is a countablex-admissible ordinal, then α is an L-cardinal.

Harrington’s proof of “Det(Σ11) implies 0] exists” in ZF is done in

two steps:

First Step Det(Σ11) implies HP;

Second Step HP implies 0] exists.

So in ZF we have:

Det(Σ11)⇔ HP⇔ 0] exists.


Analysis of Lightface Harrington’s theorem

Definition

We let Harrington’s Principle, HP for short, denote the followingstatement: there is a real x such that if α is a countablex-admissible ordinal, then α is an L-cardinal.

Harrington’s proof of “Det(Σ11) implies 0] exists” in ZF is done in

two steps:

First Step Det(Σ11) implies HP;

Second Step HP implies 0] exists.

So in ZF we have:

Det(Σ11)⇔ HP⇔ 0] exists.


Z2 + Det(Σ11) implies HP

Fact

(Z2) Det(Σ11) implies HP.

Harrington, Frideman Proof via Steel forcing

Sami Proof via effective descriptive set theory, totallyforcing-free

Woodin Proof via Barwise-compactness theorem and basicCohen forcing Col(ω, α)



Fact







Fact







Fact






Z2 + HP does not imply 0]

Question

Is “HP implies 0] exists” provable in Z2?

If yes, then lightface Harrington’s theorem is provable in Z2.

If no, then is “HP implies 0] exists” provable in Z3 or Z4?

Fact

Z2 + 0] exists implies HP.

Theorem

(Set forcing) Assuming ω1 is a limit cardinal in L, we can force aset model of Z2 + HP.

Corollary

Z2 + HP does not imply 0] exists.



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Fact


Theorem


Corollary




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Fact


Theorem


Corollary




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Fact


Theorem


Corollary



The notion of remarkable cardinal

Theorem

(Magidor) A cardinal κ is supercompact if and only if for everyregular cardinal λ > κ there is a regular cardinal λ < κ and anelementary embedding j : Hλ → Hλ with j(crit(j)) = κ.

We can view a remarkable cardinal as a type of genericsupercompact cardinal using Magidor’s theorem.

Definition

(Schindler) A cardinal κ is remarkable if for every regular cardinalλ > κ, there is a regular cardinal λ < κ such that in V Col(ω,<κ)

there is an elementary embedding j : HVλ→ HV

λ withj(crit(j)) = κ.


The notion of remarkable cardinal

Theorem

(Magidor) A cardinal κ is supercompact if and only if for everyregular cardinal λ > κ there is a regular cardinal λ < κ and anelementary embedding j : Hλ → Hλ with j(crit(j)) = κ.

We can view a remarkable cardinal as a type of genericsupercompact cardinal using Magidor’s theorem.

Definition

(Schindler) A cardinal κ is remarkable if for every regular cardinalλ > κ, there is a regular cardinal λ < κ such that in V Col(ω,<κ)

there is an elementary embedding j : HVλ→ HV

λ withj(crit(j)) = κ.



Question


Theorem

(Set forcing) Assuming there exists a remarkable cardinal with aweakly inaccessible cardinal above it, we can force a model ofZ3 + HP.

Corollary




Question


Theorem

(Set forcing) Assuming there exists a remarkable cardinal with aweakly inaccessible cardinal above it, we can force a model ofZ3 + HP.

Corollary



Strong reflecting property

Definition

Let γ ≥ ω1 be an L-cardinal. γ has the strong reflecting propertyfor L-cardinals, denoted SRPL(γ), if and only if for some regularcardinal κ > γ, if X ≺ Hκ, |X | = ω and γ ∈ X , then γ is anL-cardinal. If γ < ω1, we say that γ has the strong reflectingproperty iff γ = γ.

The first step Force a club in ω2 of L-cardinals with the strongreflecting property via set forcing.

The second step Force a set model of Z3 + HP via set forcingwithout reshaping.



Definition






Definition





Comments on the proof

The hypothesis “there exists a remarkable cardinal with aweakly inaccessible cardinal above it” is only used in the firststep.

The proof is done via set forcing without use of reshaping.

Only assuming one remarkable cardinal is not enough for ourproof.

The proof uses that SRPL(ω2). Only knowing that SRPL(γ)for some γ ∈ [ω1, ω2) is not enough for the proof.

For my proof of forcing a set model of Z3 + HP, theassumption that there exists a remarkable cardinal with aweakly inaccessible cardinal above it is optimal.





































Characterization of strong reflecting property for ωn

Theorem

(1) The following are equivalent:

(i) SRPL(ω1).(ii) ω1 is a limit cardinal in L.

(2) (Set forcing) The following two theories are equiconsistent:

(i) SRPL(ω2);(ii) ZFC + there exists a remarkable cardinal with a weakly

inaccessible cardinal above it.


(i) SRPL(γ) holds for some L-cardinal γ > ω2.(ii) 0] exists.

(iii) SRPL(γ) holds for every L-cardinal γ ≥ ω1.



Theorem











Theorem











Theorem







(i) SRPL(γ) holds for some L-cardinal γ > ω2.(ii) 0] exists.(iii) SRPL(γ) holds for every L-cardinal γ ≥ ω1.


Large cardinal strength of “Z3 + HP”

Question

What is the large cardinal strength of “Z2 + HP” and “Z3 + HP”?

Theorem (joint with Schindler)

Class forcing Z2 + HP is equiconsistent with ZFC.

Class forcing The following two theories are equiconsistent:

(1) Z3 + HP.(2) ZFC + there exists a remarkable cardinal.

The Z3 proof is done in two steps:

(1) Z3 + HP implies L |= ZFC + ωV1 is remarkable.

(2) If “ZFC + there exists a remarkable cardinal” is consistent,then “Z3 + HP” is consistent.


Large cardinal strength of “Z3 + HP”

Question

What is the large cardinal strength of “Z2 + HP” and “Z3 + HP”?

Theorem (joint with Schindler)

Class forcing Z2 + HP is equiconsistent with ZFC.

Class forcing The following two theories are equiconsistent:

(1) Z3 + HP.(2) ZFC + there exists a remarkable cardinal.

The Z3 proof is done in two steps:

(1) Z3 + HP implies L |= ZFC + ωV1 is remarkable.

(2) If “ZFC + there exists a remarkable cardinal” is consistent,then “Z3 + HP” is consistent.


Two ways to force a model of Z3 + HP

Set forcing Assuming there exists a remarkable cardinal with aweakly inaccessible cardinal above it, withoutreshaping

Class forcing Assuming there exists a remarkable cardinal, withreshaping

Question

1 Is it true that if we can force a model of proposition φ by classforcing assuming some large cardinal axioms, then we couldforce a model of φ by set forcing assuming the same largecardinal axioms?

2 Could we force a model of Z3 + HP via set forcing onlyassuming the existence of one remarkable cardinal?





Question







Question







Question




Z4 + HP implies 0]

Question


Definition

Two definitions of 0] in Z2:

(1) 0] is the unique well founded remarkable cofinal EM set;

(2) 0] is the real which codes a countable iterable premouse.

Theorem

(Z4) The following are equivalent:

(1) HP.

(2) Lω2 has an uncountable set of indiscernibles.

(3) 0] exists.


Z4 + HP implies 0]

Question


Definition




Theorem


(1) HP.


(3) 0] exists.


Z4 + HP implies 0]

Question


Definition




Theorem


(1) HP.


(3) 0] exists.


Boldface Harrington’s theorem

Fact

Both Lightface and Boldface Martin’s theorem are provable in Z2.

Theorem

Boldface Harrington’s theorem is provable in Z2.

Definition

For any real x, let HP(x) denote the statement: there is a real ysuch that if α is a countable y-admissible ordinal, then α is anL[x ]-cardinal.

The proof is done in two steps:

1 Z2 + Det(Σ11) implies for any real x , HP(x) holds;

2 Z2 + ∀x ∈ ωω(HP(x)) implies that for any real x , x ] exists.



Fact


Theorem


Definition







Fact


Theorem


Definition







Fact


Theorem


Definition







Fact


Theorem


Definition






Summary

I find an interesting classic mathematical theorem expressiblein Z2 but not provable in Z2: “HP implies 0] exists”.

“HP implies 0] exists” is not provable in Z3.

In Z4, HP is equivalent to 0] exists.

Z4 is the minimal system in higher order arithmetic to showthat HP implies 0] exists.

Z2 + HP is equiconsistent with ZFC.

Z3 + HP is equiconsistent with ZFC + there exists aremarkable cardinal.

Lightface Martin’s theorem, Boldface Martin’s theorem andBoldface Harrington’s theorem are all provable in Z2.


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Reference list

Yong Cheng, Forcing a set model of Z3 + Harrington’sPrinciple, Mathematical Logic Quarterly 61, No. 4-5, 274-287,2015.

Yong Cheng, The strong reflecting property and Harrington’sPrinciple, Mathematical Logic Quarterly, 61, No. 4-5, 329-340,2015.

Yong Cheng and Ralf Schindler, Harrington’s Principle inhigher order arithmetic, The Journal of Symbolic Logic,Volume 80,Issue 02, June 2015, pp 477-489.

Yong Cheng, Incompleteness for higher order arithmetic: Aspecific example from Set Theory, Manuscript, to appear.


Thanks for your attention!


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Incompleteness for higher order arithmetic: A speci c ...yongcheng.whu.edu.cn/webPageContent/talk/...higher-order-arithmeti… · Yong Cheng Incompleteness for higher order arithmetic: