Comparing Peano Arithmetic, Basic Law V, and Hume’s ... · Sean Walsh Department of Philosophy, Birkbeck, University of London, and the Plurals, Predicates, and Paradox Project

PA2 vs. BL2 vs. HP2 Philosophical Significance Subsystems of PA2 Subsystems of BL2 Subsystems of HP2

Comparing Peano Arithmetic, Basic Law V, and Hume’s Principle

Sean WalshDepartment of Philosophy, Birkbeck, University of London,

and the Plurals, Predicates, and Paradox Project (ERC)[email protected] or [email protected]

http://www.swalsh108.org

Logic Seminar, Goteborgs Universitet, 15 April 2011

mailto:[email protected]

mailto:[email protected]

http://www.swalsh108.org

http://www.flov.gu.se/forskning/forskningsomraden/logik/logikseminariet/


Overview

Over the past 20 years, philosophers have studied the systemsclosely related to subsystems of second-order arithmetic.

These systems are different in that they don’t have the ringlanguage but have some additional function symbols which takethe second-order part to the first-order part.

In this talk, we describe some new results on the interpretabilitystrength of these theories. In essence these results come down tothe construction of new models of these theories.

These constructions use tools from computability theory, including:hyperarithmetic theory, computable model theory & reverse math.

Part of the motivation of these results is to understand better thenature and limits of logicism in the philosophy of mathematics.


Robinson’s Q, Mathematical Induction & PA2

The axioms of Robinson’s Q are the following:

(Q1) x + 1 6= 0

(Q2) x + 1 = y + 1→ x = y

(Q3) x 6= 0→ ∃ w x = w + 1

(Q4) x + 0 = x

(Q5) x + (y + 1) = (x + y) + 1

(Q6) x · 0 = 0

(Q7) x · (y + 1) = x · y + x

(Q8) x ≤ y ↔ ∃ z x + z = y .


Robinson’s Q, Mathematical Induction & PA2

Mathematical induction or MI says that

0 ∈ X & ∀ x(x ∈ X → Sx ∈ X )]→ ∀x x ∈ X

The theory PA2= Q + MI + the comprehension schema:

∃ R [∀ x x ∈ R ↔ ϕ(x)]

The standard model of PA2 is

N2 = (ω,P(ω), 0, 1,+,×,≤,∈)


The Language of PA2

The language of Robinson’s Q is the natural language for

N = (ω, 0, 1,+,×,≤)

namely the language of ordered rings.

The language of PA2 is the natural language for the structure

N2 = (ω,P(ω), 0, 1,+,×,≤,∈)

namely a two-sorted language with a relation ∈ between the sorts.

Often x ∈ X is written as Xx , and so we might suppress the ∈.


The Language of PA2


N = (ω, 0, 1,+,×,≤)



N2 = (ω,P(ω), 0, 1,+,×,≤,∈)




The Language of PA2


N = (ω, 0, 1,+,×,≤)



N2 = (ω,P(ω), 0, 1,+,×,≤,∈)




An ω-sorted LanguageMore generally, we can consider a language for structures

(M,S1, S2, . . . ,∈1,∈2, . . .)

where Sn ⊆ P(Mn)

and where ∈n holds between n-tuples from M and elements of Sn.

Often (a, b) ∈2 R is written Rab, and so we might suppress ∈n.

Natural structures in this language include

(M,P(M),P(M2), . . .)

where M is any set and

(M,D1(M),D2(M), . . .)

where Dn(M) ⊆ Mn are the definable subsets of structure M


An ω-sorted LanguageMore generally, we can consider a language for structures

(M,S1, S2, . . . ,∈1,∈2, . . .)

where Sn ⊆ P(Mn)

and where ∈n holds between n-tuples from M and elements of Sn.

Often (a, b) ∈2 R is written Rab, and so we might suppress ∈n.

Natural structures in this language include

(M,P(M),P(M2), . . .)

where M is any set and

(M,D1(M),D2(M), . . .)

where Dn(M) ⊆ Mn are the definable subsets of structure M


The Languages of BL2 & HP2

The languages of BL2 and HP2 describe structures

(M,S1,S2, . . . ,#)

where # : S1 → M is a function.

That is, their languages add function symbols between the sorts.

So the languages of BL2 and HP2 naturally generalize that of PA2:

I First we delete the ring language

I Second we add more sorts for sets of e.g. ordered pairs

I Third we add function symbols between the sorts


The Languages of BL2 & HP2

The languages of BL2 and HP2 describe structures

(M,S1,S2, . . . ,#)

where # : S1 → M is a function.

That is, their languages add function symbols between the sorts.

So the languages of BL2 and HP2 naturally generalize that of PA2:

I First we delete the ring language

I Second we add more sorts for sets of e.g. ordered pairs

I Third we add function symbols between the sorts


Hume’s Principle & HP2

Hume’s Principle is the following sentence in the language of HP2:

#X = #Y ⇐⇒ ∃ bijection f : X → Y

The theory HP2 is Hume’s Principle & the comprehension schema:

∃ R [∀ x x ∈ R ↔ ϕ(x)]

Natural models of HP2 include

(α,P(α),P(α2), . . . ,#)

where α is an ordinal which is not a cardinal and #X = |X |


Hume’s Principle & HP2

Hume’s Principle is the following sentence in the language of HP2:

#X = #Y ⇐⇒ ∃ bijection f : X → Y

The theory HP2 is Hume’s Principle & the comprehension schema:

∃ R [∀ x x ∈ R ↔ ϕ(x)]

Natural models of HP2 include

(α,P(α),P(α2), . . . ,#)

where α is an ordinal which is not a cardinal and #X = |X |


Basic Law V & BL2

Basic Law V is the following sentence in the language of BL2

#X = #Y ⇐⇒ X = Y ⇐⇒ ∀ z (Xz ↔ Yz)

The theory BL2 is Basic Law V & the comprehension schema:

∃ R [∀ x x ∈ R ↔ ϕ(x)]


Basic Law V & BL2


∂X = ∂Y ⇐⇒ X = Y ⇐⇒ ∀ z (Xz ↔ Yz)


∃ R [∀ x x ∈ R ↔ ϕ(x)]

Russell’s paradox shows us that BL2 is inconsistent:

∃ X [∀ x x ∈ X ↔ (∃ Y ∂Y = x & x /∈ Y )]

Then

∂X ∈ X ↔ ∂X /∈ X


Basic Law V & BL2


∂X = ∂Y ⇐⇒ X = Y ⇐⇒ ∀ z (Xz ↔ Yz)


∃ R [∀ x x ∈ R ↔ ϕ(x)]

Russell’s paradox shows us that BL2 is inconsistent:

∃ X [∀ x x ∈ X ↔ (∃ Y ∂Y = x & x /∈ Y )]

Then

∂X ∈ X ↔ ∂X /∈ X


Subsystems of PA2, BL2, HP2

Let us temporarily rename PA2 as CA2.

Suppose that XY2 is one of CA2, BL2, or HP2. Then we define:

AXY0 = restriction of XY2 to arithmetical comprehension schema

∆11 − XY0 = restriction of XY2 to ∆11-comprehension schema:

[∀ n ϕ(n)↔ ψ(n)]→ [∃ X ∀ n n ∈ X ↔ ϕ(n)],where ϕ Σ11 & ψ Π1

1

Σ11 − YX0 = AXY0 and Σ11-choice schema:

[∀ n ∃ X ϕ(n,X )]→ [∃ R ∀ n ϕ(n,Rn)],where ϕ Σ11

Π11 − XY0 = restriction of XY2 to Π11-comprehension schema


Proposition

(Simpson [15] p. 298) Σ11 − YX0 ⇒ ∆11 − XY0.

Proof.Let M = (M,S1, S2, . . .) be a model of Σ11 − YX0.Suppose that M |= ∀ x ϕ(x)↔ ψ(x), where ϕ is Σ1

1 and ψ is Π11.

Then M |= ∀ x ϕ(x) ∨ ¬ψ(x). Fix a ∈ M. Then

M |= ∀ x ∃ Z (ϕ(x) ∧ a ∈ Z ) ∨ (¬ψ(x) ∧ a /∈ Z )

By the Σ11-choice schema, there is R such that

M |= ∀ x (ϕ(x) ∧ a ∈ Rx) ∨ (¬ψ(x) ∧ a /∈ Rx)

By AXY0, there is W such that x ∈W if and only if a ∈ Rx .Then x ∈W if and only if ϕ(x).


The Provability Relation

The previous proposition tells us that the red theories prove theblack theories immediately below them.

Π11 − BL0

��

Π11 − CA0

��Σ11 − LB0

��

Σ11 − AC0

��

Π11 − HP0

%%LLLLLLLLLLΣ11 − PH0

yyrrrrrrrrrr

∆11 − BL0

��

∆11 − CA0

��

∆11 − HP0

��ABL0 ACA0 AHP0



The Russell paradox is derivable in Π11 − BL0 (since the set inquestion is Σ1

1-definable) so we remove it from the diagram.

Π11 − BL0

��

Π11 − CA0

��Σ11 − LB0

��

Σ11 − AC0

��

Π11 − HP0


yyrrrrrrrrrr

∆11 − BL0

��

∆11 − CA0

��

∆11 − HP0




The Russell paradox is derivable in Π11 − BL0 (since the set inquestion is Σ1

1-definable) so we remove it from the diagram.

Π11 − CA0

��Σ11 − LB0

��

Σ11 − AC0

��

Π11 − HP0


yyrrrrrrrrrr

∆11 − BL0

��

∆11 − CA0

��

∆11 − HP0




It is not obvious from the definitions, but one can in fact show thatΠ11 − CA0 implies Σ11 − AC0 (see [15] Theorem V.8.3 p. 205).

Π11 − CA0

��Σ11 − LB0

��

Σ11 − AC0

��

Π11 − HP0


yyrrrrrrrrrr

∆11 − BL0

��

∆11 − CA0

��

∆11 − HP0




So here are the provability implications which we have now:

Π11 − CA0

��Σ11 − LB0

��

Σ11 − AC0

��

Π11 − HP0


yyrrrrrrrrrr

∆11 − BL0

��

∆11 − CA0

��

∆11 − HP0




In fact we can show the following, where ⇒ is irreversibleimplication, → is reversal not known, and 9 is non-implication

Π11 − CA0

��Σ11 − LB0

��

Σ11 − AC0

��

Π11 − HP0

!)LLLLLLLLL

LLLLLLLLL

?--Σ11 − PH0|mm

u} rrrrrrrrr

rrrrrrrrr

∆11 − BL0

��

∆11 − CA0

��

∆11 − HP0



The Intepretability RelationAnother important relationship is that of interpretability:

Π11 − CA0

��u} rrrrrrrrr

rrrrrrrrroo // Π11 − HP0

Σ11 − LB+0

--

��

Σ11 − AC0?mm

��Σ11 − LB0

��

∆11 − CA0

��∆11 − BL0

��

ACA0

��ABL0

t|

rrrrrrrrrrr

rrrrrrrrrrrQ//oo ? --

Σ11 − PH0jj

!)

LLLLLLLLLL

LLLLLLLLLL


The Intepretability Relation

The relation of interpretation between theories is a uniform versionof the relation of definability between structures.

Recall that M′ is interpretable in M (written M′ ≤I M) if someisomorphic copy of M′ is definable in M.

Semantically, T ′ is interpretable in T (written T ′ ≤I T ) if everymodel M of T uniformly interprets without parameters somemodel M′ of T ′, where uniformly means that the same formulasare used each time.

Syntactically, T ′ is interpretable in T (written T ′ ≤I T ) if there isa translation of the primitives of T ′ into formulas of T such thatthe translations of T ′-theorems are T -theorems.

E.g. PA2 ≤I ZFC or groups ≤I fields.



Typically the only way to prove ≤I is to find an interpretation.

To prove �I one can use the following proposition:

Proposition

If ACA0 ⊆ T ⊆ PA2 is finitely axiomtizable and T proves Con(S),where S is a computable theory, then T �I S.

Typically the way we use this is as follows:

We take ACA0 ⊆ T ⊆ PA2 and show T proves Con(S).

Our proof in fact typically shows that S ≤I T .

So we conclude that S <I T .



Frege’s project (cf. [10]) was to interpret PA2 within BL2 via HP2:

BL2 // HP2 // PA2



Again, because its inconsistent, we remove BL2 from the diagram:

BL2 // HP2 // PA2



Again, because its inconsistent, we remove BL2 from the diagram:

HP2 // PA2



Boolos [2] noted that PA2 interprets HP2:

HP2 // PA2



Boolos [2] noted that PA2 interprets HP2:

HP2 oo // PA2



Linnebo [13] noted this was uniform in the levels comprehension.

HP2 oo // PA2




Π1n − HP0oo // Π1n − CA0




Π11 − HP0oo // Π11 − CA0




Π11 − CA0oo // Π11 − HP0



Since interpretability respects provability, we can add:

Π11 − CA0oo // Π11 − HP0


The Intepretability RelationSince interpretability respects provability, we can add:

Π11 − CA0oo //

��

Π11 − HP0

��

Σ11 − LB0

��

Σ11 − AC0

��

Σ11 − PH0

yyrrrrrrrrrr

∆11 − BL0

��

∆11 − CA0

��

∆11 − HP0



The Intepretability RelationHeck [11] showed ABL0 interprets Q and Burgess [4] did so for AHP0:

Π11 − CA0oo //

��

Π11 − HP0

��

Σ11 − LB0

��

Σ11 − AC0

��

Σ11 − PH0

yyrrrrrrrrrr

∆11 − BL0

��

∆11 − CA0

��

∆11 − HP0



The Intepretability RelationHeck [11] showed ABL0 interprets Q and Burgess [4] did so for AHP0:

Π11 − CA0oo //

��

Π11 − HP0

��

Σ11 − LB0

��

Σ11 − AC0

��

Σ11 − PH0

yyrrrrrrrrrr

∆11 − BL0

��

∆11 − CA0

��

∆11 − HP0

��ABL0

&&NNNNNNNNNNNN ACA0

��

AHP0

xxpppppppppppp

Q


The Intepretability RelationGanea [9] and Visser [18] showed that Q interprets ABL0:

Π11 − CA0oo //

��

Π11 − HP0

��

Σ11 − LB0

��

Σ11 − AC0

��

Σ11 − PH0

yyrrrrrrrrrr

∆11 − BL0

��

∆11 − CA0

��

∆11 − HP0

��ABL0

&&NNNNNNNNNNNN ACA0

��

AHP0

xxpppppppppppp

Q


The Intepretability RelationGanea [9] and Visser [18] showed that Q interprets ABL0:

Π11 − CA0oo //

��

Π11 − HP0

��

Σ11 − LB0

��

Σ11 − AC0

��

Σ11 − PH0

yyrrrrrrrrrr

∆11 − BL0

��

∆11 − CA0

��

∆11 − HP0

��ACA0

��

AHP0

xxqqqqqqqqqqqq

ABL0oo // Q


The Intepretability RelationFerreira & Wehmeier [6] show Π11 − CA0 implies Con(∆11 − BL0):

Π11 − CA0oo //

��

Π11 − HP0

��

Σ11 − LB0

��

Σ11 − AC0

��

Σ11 − PH0

yyrrrrrrrrrr

∆11 − BL0

��

∆11 − CA0

��

∆11 − HP0

��ACA0

��

AHP0

xxqqqqqqqqqqqq

ABL0oo // Q


The Intepretability RelationFerreira & Wehmeier [6] show Π11 − CA0 implies Con(∆11 − BL0):

Π11 − CA0oo //

��

��Π11 − HP0

��

Σ11 − LB0

��

Σ11 − AC0

��

Σ11 − PH0

yyrrrrrrrrrr

∆11 − BL0

��

∆11 − CA0

��

∆11 − HP0

��ACA0

��

AHP0

xxqqqqqqqqqqqq

ABL0oo // Q


The Intepretability RelationA slight modification shows Π11 − CA0 implies Con(Σ11 − LB0):

Π11 − CA0oo //

��

��Π11 − HP0

��

Σ11 − LB0

��

Σ11 − AC0

��

Σ11 − PH0

yyrrrrrrrrrr

∆11 − BL0

��

∆11 − CA0

��

∆11 − HP0

��ACA0

��

AHP0

xxqqqqqqqqqqqq

ABL0oo // Q


The Intepretability RelationA slight modification shows Π11 − CA0 implies Con(Σ11 − LB0):

Π11 − CA0oo //

��yyrrrrrrrrrrΠ11 − HP0

��

Σ11 − LB0

��

Σ11 − AC0

��

Σ11 − PH0

yyrrrrrrrrrr

∆11 − BL0

��

∆11 − CA0

��

∆11 − HP0

��ACA0

��

AHP0

xxqqqqqqqqqqqq

ABL0oo // Q


The Intepretability RelationA new proof shows there is theory Σ11 − LB

+0 such that:

Π11 − CA0oo //


��

Σ11 − LB+0

��

--Σ11 − AC0

��

Σ11 − PH0

yyssssssssss

Σ11 − LB0

��

∆11 − CA0

��

∆11 − HP0

��∆11 − BL0

��

ACA0

��

AHP0

xxqqqqqqqqqqqq

ABL0oo // Q


The Intepretability RelationA new proof shows there is theory Σ11 − LB

+0 such that:

Π11 − CA0oo //


��

Σ11 − LB+0

��

--Σ11 − AC0

��

Σ11 − PH0

yyssssssssss

Σ11 − LB0

��

∆11 − CA0

��

∆11 − HP0

��∆11 − BL0

��

ACA0

��

AHP0

xxqqqqqqqqqqqq

ABL0oo // Q


The Intepretability RelationAnother new proof shows that ACA0 proves Con(Σ11 − PH0):

Π11 − CA0oo //


��

Σ11 − LB+0

��

--Σ11 − AC0

��

Σ11 − PH0

yyssssssssss

Σ11 − LB0

��

∆11 − CA0

��

∆11 − HP0

��∆11 − BL0

��

ACA0

��

AHP0

xxqqqqqqqqqqqq

ABL0oo // Q


The Intepretability RelationAnother new proof shows that ACA0 proves Con(Σ11 − PH0):

Π11 − CA0oo //


Σ11 − LB+0

��

--Σ11 − AC0

��Σ11 − LB0

��

∆11 − CA0

��∆11 − BL0

��

ACA0

�� %%LLLLLLLLLLL

ABL0oo // Q oo Σ11 − PH0


The Intepretability RelationSince the arrows we get from proving consistency are irreversible:

Π11 − CA0oo //


Σ11 − LB+0

��

--Σ11 − AC0

��Σ11 − LB0

��

∆11 − CA0

��∆11 − BL0

��

ACA0

�� %%LLLLLLLLLLL



The Intepretability RelationSince the arrows we get from proving consistency are irreversible:

Π11 − CA0oo //

��u} rrrrrrrrr

rrrrrrrrrΠ11 − HP0

Σ11 − LB+0

��

--Σ11 − AC0

��Σ11 − LB0

��

∆11 − CA0

��∆11 − BL0

��

ACA0

��t| rrrrrrrrrrr

rrrrrrrrrrr

!)LLLLLLLLLL

LLLLLLLLLL




Π11 − CA0oo //

��u} rrrrrrrrr

rrrrrrrrrΠ11 − HP0

Σ11 − LB+0

��

--Σ11 − AC0

��Σ11 − LB0

��

∆11 − CA0

��∆11 − BL0

��

ACA0

��t| rrrrrrrrrrr

rrrrrrrrrrr

!)LLLLLLLLLL

LLLLLLLLLL



The Intepretability RelationAdding credits and the some of the open questions:

Π11 − CA0

��u} rrrrrrrrr

rrrrrrrrrooFrege/Boolos/Linnebo// Π11 − HP0

Σ11 − LB+0

Walsh --

��

Σ11 − AC0?mm

��Σ11 − LB0

��

∆11 − CA0

��∆11 − BL0

��

ACA0

��ABL0

t|

rrrrrrrrrrr

rrrrrrrrrrrQ//

Heck/Ganea/Visseroo ? --

Σ11 − PH0Burgess

jj!)

WalshLLLLLLLLLL

LLLLLLLLLL


Logicism about Arithmetical Knowledge

The logicist’ idea is that arithmetical knowledge is grounded inlogical knowledge, viz. Frege on mathematical induction:

“One will be able to see from this essay that even inferences whichare apparently particular to mathematics, like the inference from nto n + 1, are based on general logical laws, so that they do notrequire particular laws of aggregative thought” ([7] p. iv, cf. § 80p. 93, § 108 p. 118, [8] p. 104)

Wright is an important contemporary advocate of logicism:

“Anyone who accepts the Peano axioms as truths ‘not of ourmaking’ must recognise the question of what account should begiven of our ability to apprehend their truth. If Frege’s attempt toground that apprehension in pure logic were to succeed, we shouldhave an answer [. . . ] ” ([19] p. 131, cf. p. xiv).


Two Central Tasks of Logicism

The logicist account thus attempts to reduce our knowledge ofmathematical induction and the Peano axioms to our knowledge ofmore basic truths whose logical credentials are more apparent.

The logicist is then obliged to provide us with

(i) an account of our knowledge of these basic truths,

(ii) an account of the reduction which mediates our inference fromknowledge of these basic truths to knowledge of the Peano axioms.

. . . Given Frege’s result that PA2 is interpretable in HP2, it is thusnatural to take the notion of reduction from (ii) to be that ofinterpretability. This suggests the following version of the logicistargument.


The Logicist Template

This suggests the following version of the logicist argument:

The Logicist Template

Base Premise: Hume’s principle is known.

Interpretability Premise: It is known that the Peano axiomsare interpretable in Hume’s Principle.

Preservation Premise: If it is known that principles P areinterpretable in principles P∗, and principles P∗ are known,then principles P are known.

Conclusion: The Peano axioms are known.


Technical Questions Motivated by The Logicist TemplateI wanted to understand how flexible the interpretability premise is.

Interpretability Premise: It is known that the Peano axiomsare interpretable in Hume’s Principle.

In particular, I wanted to know:

1. Is this premise sensitive to the difference between predicativeand impredicative comprehension? Is there predicative versionof Frege’s result that PA2 is interpretable in HP2?

If predicative means ∆11, then answer is “no.”

2. Can Hume’s Principle be replaced by a consistent subsystemof Basic Law V?

This is still open. I conjecture: answer is “no.” Today Ifocus on preliminary step of building models of Basic Law Vwhich are mutually interpretable with the natural numbers.


Three Constructions of Models of ∆11 − CA0

Recall ∆11 − CA0 is PA2 restricted to the comprehension schema:

[∀ n ϕ(n)↔ ψ(n)]→ [∃ X ∀ n n ∈ X ↔ ϕ(n)]

where ϕ is Σ11 and ψ is Π1

1.

There are three constructions of models of ∆11 − CA0:

I Kleene’s [12] proof using hyperarithmetical sets

I Barwise & Schlipf’s [1] recursively saturated models

I Steel’s construction [16] using forcing with tagged trees


Kleene’s Construction: Hyperarithmetic Sets

Kleene defined HYP and proved that (ω,HYP) |= ∆11 − CA0.

Just like Y recursive ⇔ Y ≤T ∅,

so Y ∈ HYP⇔ Y ∈ HYP∅ ⇔ Y ≤h ∅

There are a couple equivalent characterizations of HYP:

Proposition

Suppose that X ,Y ∈ 2ω. Then TFAE:

I Y ∈ HYPX or Y ≤h X

I Y is ∆11-definable over the model (ω,P(ω),X )

I There is a ∈ OX and e ∈ ω such that Y = ϕHX

ae

I Y ∈ LωCK1

[X ], where Lα[X ] is Godel’s L up to α relative to X .


Kleene’s Construction: Computable Ordinals

Suppose that X ∈ 2ω and α < ω1. Then α is X -computable if(α,∈) has an X -computable isomorphic copy (M,≺).

Let ωX1 be the least non-X -computable ordinal and let ωCK

1 be theleast non-computable ordinal.

If α is X -computable then α + 1 is X -computable, andif αn is uniformly X -computable where αn < αn+1

then supn αn is X -computable.


Kleene’s Construction: Kleene’s O

Kleene’s OX is a Π1,X1 -real consisting of “codes” for all

the X -computable ordinals. So we define simultaneously

the set OX , partial order < on OX and a decoding |·| : OX → ωX1

1 7→ 0

2a 7→ |a|+ 1

3 · 5e 7→ supn

∣∣∣ϕXe (n)

∣∣∣where ϕX

e (n) < ϕXe (n + 1)


Kleene’s Construction: The H-sets

The sets HXa for a ∈ OX are defined as follows:

HX1 = X

HX2a = (HX

a )′

HX3·5e = ⊕nHX

ϕXe (n)

What Kleene showed was that the following are equivalent:

I Y is ∆11-definable over the model (ω,P(ω),X )

I There is a ∈ OX and e ∈ ω such that Y = ϕHX

ae


Kleene’s Construction: The Key Theorem

Kleene [12] proved the following important theorem:

TheoremThe predicate ∃ Y ≤h X ϕ(Y ,X ) where ϕ is Π1

1 is a Π11-predicate.

There is a converse to this, called the Spector-Gandy theorem:

TheoremIf ψ(X ) is a Π1

1-predicate, then there is arithmetical ϕ(Y ,X )

ψ(X )⇔ ∃ Y ≤h X ϕ(Y ,X )

See Sacks [14] p. 61 for proof and references.


Kleene’s Construction

Theorem(Kleene) M = (ω,HYP) is a model of ∆11 − CA0.

Proof.Suppose that M |= ∀ n ∃ Y ϕ(Y , n)↔ ¬∃ Y ψ(Y , n)

where ϕ and ψ are arithmetical.

Then for all n ∈ ω

∃ Y ≤h ∅ ϕ(Y , n)⇔ ¬∃ Y ≤h ∅ ψ(Y , n)

By the previous proposition, the LHS is Π11 and the RHS is Σ1

1.

Hence, the set of X = {n : ∃ Y ≤h ∅ ϕ(Y , n)} is ∆11

and so by Kleene’s result X ∈ HYP.


The Barwise-Schlipf ConstructionBarwise & Schlipf found another way to build models of ∆11 − CA0.

Their construction is based on idea of recursively saturated model.

A countable model M is X -recursively saturated if everyX -recursive partial type p(v) over a finite number of parametersis realized in the model.

Proposition

For every countable model M and every X ∈ 2ω, there is acountable X -recursively saturated elementary extension N �M

This proposition is in fact provable in WKL0 (cf. [15] p. 383).


The Barwise-Schlipf Construction

Suppose N is recursively saturated and θi is computable. Then:

[∀ K > 0 N |= ∃ nK∧

i=1

¬θi (n)] =⇒ [N |= ∃ n∧i

¬θi (n)]

Note that the contrapositive is the following:

[N |= ∀ n∨i

θi (n)] =⇒ [∃ K > 0 N |= ∀ nK∨

i=1

θi (n)]


The Barwise-Schlipf Construction

TheoremSuppose that N is a recursively saturated model of PA.Then M = (N,D1(N )) is a model of ∆11 − CA0,where D1(N ) are the definable subsets of N .

TheoremSuppose that M |= ∀ n ∃ Y ϕ(Y , n)↔ ¬∃ Y ψ(Y , n)

where ϕ and ψ are arithmetical. Then

N |= ∀ n∨θ

∃ b ϕ(θ(·, b), n) ∨ ¬ψ(θ(·, b), n)

Then there are θ1, . . . , θK such that

N |= ∀ nK∨

i=1

∃ b ϕ(θi (·, b), n) ∨ ¬ψ(θi (·, b), n)


Ferreira & Wehmeier’s Barwise-Schlipf Model of Σ11 − LB0

The first to find models of ∆11 − BL0 were Ferreira & Wehmeier [6].

By slightly modifying their construction we get model of Σ11 − LB0

Their idea is to use a Barwise-Schlipf construction.

So, recall we need to satisfy Σ11-choice schema and Basic Law V:

∂(X ) = ∂(Y )⇐⇒ X = Y ⇐⇒ ∀ z Xz ↔ Yz


Ferreira & Wehmeier’s Barwise-Schlipf Model of Σ11 − LB0

The first step is to build a sequence of Mi in Li as follows:

At stage s = 0 we let M0 be an countable structure.

At odd stages s = 2e + 1 we add skolem functions to our model.

At even stages s = 2e we add new function symbols ∂sϕ : Mn → M

where n is the number of parameter variables in ϕ, so that

Ms |= ∂sϕ(a) = ∂t

ψ(b)↔ ∀ z ϕ(z , a)↔ ψ(z , b)

Then we let Mω =⋃

sMs in signature Lω =⋃

s Ls


Ferreira & Wehmeier’s Barwise-Schlipf Model of Σ11 − LB0The second step is to take an recursively saturated N �Mω,

and let M = (N,D1(N ),D2(N ), . . . , ∂) where ∂(θ(·, a)) = ∂sθ(a).

Suppose that M |= ∀ n ∃ X ϕ(n,X , ∂(X )) where ϕ is arithmetical.

N |= ∀ n∨θ

∃ b ϕ(n, θ(·, b), ∂sθ(b))


N |= ∀ nK∨

i=1

∃ b ϕ(n, θi (·, b), ∂sθi

(b))

Since we have definable skolem functions, there is definable f

N |= ∀ nK∨

i=1

ϕ(n, θi (·, f (n)), ∂sθi

(f (n)))


A Kleene Model of Σ11 − LB0

The advantage of Ferreira-Wehmeier construction is its generality.

The disadvantage is that it is less than clear what you end up with.

Our goal now is to produce a Kleene-Model of Σ11 − LB0.

That is, we want our model to have the form:

(ω,HYP, ∂), where ∂ : HYP ↪→ ω



As a first step let us consider the relation

Q(X , a, e) ≡ X ∈ HYP & a ∈ O & e ∈ ω & X = ϕHae

After recalling some definitions, we see that Q is a Π11-relation.

By Π11-uniformization, there is Π1

1-relation q such that

q(X , a, e) =⇒ Q(X , a, e)

∃ a, e Q(X , a, e) =⇒ ∃ ! a, e q(X , a, e)

Then define ∂ : HYP ↪→ ω by ∂(X ) = 〈a, e〉 iff q(X , a, e)

Note by Spector-Gandy that graph(∂) is definable over (ω,HYP).



Now as a second step we show M |= (ω,HYP, ∂) |= ∆11 − BL0.

Suppose that M |= ∀ n ∃ Y ϕ(Y , n, ∂(Y ))↔ ¬∃ Y ψ(Y , n, ∂(Y ))

where ϕ and ψ are arithmetical.

Then for all n ∈ ω

∃ Y ∈ HYP (∃ e ∂(Y ) = e & ϕ(Y , n, e))

⇐⇒ ¬∃ Y ∈ HYP (∃ e ∂(Y ) = e & ψ(Y , n, e))

By Kleene’s Theorem, the top is Π11 and the bottom is Σ1

1.

Hence, this set is ∆11 and so by Kleene’s result is in HYP.



So we just built M = (ω,HYP, ∂) ≤I (ω, 0, 1,+,×,≤,HYP).

Can we recover the ring structure from just ∂?

Consider the function S : ω → ω given by S(x) = ∂({x}).

By ∆11 − BL0, the graph of this function S is in HYP.

Now, outside of the structure, recursively define

f (0) = ∂(∅) & f (x + 1) = S(f (x))

The graph(f ) ≤T S and so graph(f ), rng(f ) ∈ HYP.

Let N = rng(f ). Then we have:

M |= ∀ X [∂(∅) ∈ X & x ∈ X → Sx ∈ X ]→ N ⊆ X



So we showed the consistency of Σ11 − LB0 and the following axiom:

“There is a set N such that

I ∂(∅) ∈ N and ∀ x x ∈ N → ∂({x}) ∈ N

I ∀ X [∂(∅) ∈ X & x ∈ X → ∂({x}) ∈ X ]→ N ⊆ X

I There are functions ⊕, ⊗ and relations � on N such that

(N, ∂(∅), ∂({·}),⊕,⊗,�) |= Robinson’s Q”

Let Σ11 − LB0+

= Σ11 − LB0 plus this axiom.

Then we have Σ11 − AC0 ≤I Σ11 − LB0

+<I Π

11 − CA0.


Frege’s Interpretation of PA2 in HP2

Consider the structure (κ+ 1,P(κ+ 1),P((κ+ 1)2), . . . ,#)

where κ is an infinite cardinal and #(X ) = |X |. This models HP2.

Can we interpret a model of PA2 in it?

Well, we can pick out ω in this structure:

It is the # of the smallest set containing #∅

and closed under successor:

S#X = #Y ⇐⇒ ∃ y ∈ Y \ X (Y = X ∪ {x})

This is the idea behind Frege’s proof that PA2 ≤I HP2.


Boolos’ Interpretation in HP2 in PA2

Boolos [2] noted that HP2 is interpretable in PA2:

Working in PA2 we can define the # function by

#X =

{0 if X infinite,

|X |+ 1 otherwise.

This gives us the following information about provability:

Π11 − HP0

!)LLLLLLLLL

LLLLLLLLL

?--Σ11 − PH0|mm

u} rrrrrrrrr

rrrrrrrrr

∆11 − HP0

��AHP0





#X =

{0 if X infinite,

|X |+ 1 otherwise.

This gives us the following information about provability:

Π11 − HP0

!)LLLLLLLLL

LLLLLLLLL

?--Σ11 − PH0|mm

u} rrrrrrrrr

rrrrrrrrr

∆11 − HP0

��AHP0





#X =

{0 if X infinite,

|X |+ 1 otherwise.

And this gives us the following information about interpretability:

Σ11 − AC0

��

--Σ11 − PH0

��∆11 − CA0

��

--∆11 − HP0

��ACA0

,, AHP0


A Barwise-Schlipf Model of Σ11 − PH0

So let R be a countable real-closed field, with the ordering.

Then R is o-minimal:

every definable subset of R is a finite union of points and intervals.

Moreover, R has built-in bijective invariants, which are given by

dimension & Euler characteristic

To define these we need to define the notion of a cell.



Suppose that X is a definable subset of Rn.

Then C (X ) = definable continuous functions on X

& C∞(X ) = C (X ) plus −∞, +∞ as functions

& (f , g)X = {(x , r) ∈ X × R : f (x) < r < g(x)} if f < g on X

Then σ-cells for σ ∈ 2<ω are defined inductively as follows:

0-cells are points

1-cells are open intervals, including (−∞, a), (a,−∞).

σ0-cells are graphs of functions f ∈ C (X ) where X is a σ-cell

σ1-cells are sets (f , g)X where f , g ∈ C∞(X ) and X is a σ-cell


A Barwise-Schlipf Model of Σ11 − PH0For each m, we inductively define a decomposition of Rm.

A decomposition of R1 has the following form:

{(−∞, a1), (a1, a2), . . . , (ak ,+∞), {a1}, . . . , {ak}}

where a1 < a2 < · · · < ak

A decomposition of Rm+1 = Rm × R is a finite partition of Rm+1

into cells A such that the set of projections π(A) is adecomposition of Rm.

Theorem(Cell Decomposition) For any finite sequence of B-definable setsA1, . . . ,Ak ⊆ Rm, there is a decomposition of Rm partitioningeach of the Ai . Moreover, the cells in the decomposition areB-definable.



The dimension of a definable set is defined as follows:

dim(X ) = max{i1 + · · ·+ in : X contains a (i1, . . . , in)-cell}

For example X ⊆ R1, then dim(X ) > 0 iff X contains an interval.

Note that X ⊆ Rn implies that dim(X ) ≤ n.

The Euler characteristic of a definable set is defined as follows:

E (X ) = k0 − k1 + k2 − k3 · · ·

where kd is the number of d-dimensional cells in cell-decomp. of X



Dimension and Euler Characteristic are bounded & definable:

Suppose that ϕ(x , y) is a formula.

Then there is an integer Nϕ > 0 such that for all b∣∣E (ϕ(·, b))∣∣ ≤ Nϕ

and for each integer n, the following sets are definable

{b : dim(ϕ(·, b)) = n} {b : E (ϕ(·, b)) = n}

Such integers & formulas can be computed from Th(R) & ϕ.



TheoremSuppose that R is an o-minimal expansion of a real closed field.

Suppose that X ⊆ Rn and Y ⊆ Rm are definable.

Then there is a definable bijection f : X → Y if and only if

dim(X ) = dim(Y ) and E (X ) = E (Y ).

See van den Dries [17] p. 132 for proof.

So now we see how to build our model of Σ11 − PH0:

We choose a recursively saturated real closed field R.

We define M = (R,D1(R),D2(R), . . . ,#)

where #X = 〈dim(X ),E (X )〉

Then by the theorem, M models Hume’s Principle.



TheoremSuppose that R is an o-minimal expansion of a real closed field.

Suppose that X ⊆ Rn and Y ⊆ Rm are definable.

Then there is a definable bijection f : X → Y if and only if

dim(X ) = dim(Y ) and E (X ) = E (Y ).

See van den Dries [17] p. 132 for proof.

So now we see how to build our model of Σ11 − PH0:

We choose a recursively saturated real closed field R.

We define M = (R,D1(R),D2(R), . . . ,#)

where #X = 〈dim(X ),E (X )〉

Then by the theorem, M models Hume’s Principle.


A Barwise-Schlipf Model of Σ11 − PH0SupposeM |= ∀ n ∃ Y ϕ(n,Y ,#Y ) where ϕ is arithmetical. Then

R |= ∀ n∨θ

∃ b ϕ(n, θ(·, b), 〈dim(θ(·, b)),E (θ(·, b))〉)

Since dim(X ) & E (X ) are bounded & definable, these are formulas.


R |= ∀ nK∨

i=1

∃ b ϕ(n, θi (·, b), 〈dim(θi (·, b)),E (θi (·, b))〉)

Since R has definable skolem functions there is a definable f

R |= ∀ nK∨

i=1

ϕ(n, θi (·, f (n)), 〈dim(θi (·, f (n))),E (θi (·, f (n)))〉)


Summary of Main Results

In essence, we have presented two new constructions:

I A Kleene model of Σ11 − LB0 using hyperarithmetic theory

I A Barwise-Schlipf model of Σ11 − PH0 using o-minimality

Formalizing these constructions gives bounds on intepretability:

I Σ11 − AC0 ≤I Σ11 − LB0

+<I Π

11 − CA0.

I Σ11 − PH0 <I ACA0.



Π11 − CA0

��u} rrrrrrrrr

rrrrrrrrroo // Π11 − HP0

Σ11 − LB+0

--

��

Σ11 − AC0?mm

��Σ11 − LB0

��

∆11 − CA0

��∆11 − BL0

��

ACA0

��ABL0

t|

rrrrrrrrrrr

rrrrrrrrrrrQ//oo ? --

Σ11 − PH0jj

!)

LLLLLLLLLL

LLLLLLLLLL



Π11 − CA0

��Σ11 − LB0

��

Σ11 − AC0

��

Π11 − HP0

!)LLLLLLLLL

LLLLLLLLL

?--Σ11 − PH0|mm

u} rrrrrrrrr

rrrrrrrrr

∆11 − BL0

��

∆11 − CA0

��

∆11 − HP0



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[3] George Boolos.Logic, Logic, and Logic.Harvard University Press, Cambridge, MA, 1998.Edited by Richard Jeffrey.

[4] John P. Burgess.Fixing Frege.Princeton Monographs in Philosophy. PrincetonUniversity Press, Princeton, 2005.

[5] William Demopoulos, editor.Frege’s Philosophy of Mathematics.Harvard University Press, Cambridge, 1995.

[6] Fernando Ferreira and Kai F. Wehmeier.

On the Consistency of the ∆11-CA fragment of

Frege’s Grundgesetze.Journal of Philosophical Logic, 31(4):301–311,2002.

[7] Gottlob Frege.Die Grundlagen der Arithmetik.Koebner, Breslau, 1884.

[8] Gottlob Frege.Kleine Schriften.Olms, Hildesheim, 1967.Edited by Ignacio Angelelli.

[9] Mihai Ganea.Burgess’ PV is Robinson’s Q.Journal of Symbolic Logic, 72(2):618–624, 2007.

[10] Richard G. Heck, Jr.The Development of Arithmetic in Frege’sGrundgesetze der Arithmetik.The Journal of Symbolic Logic, 58(2):579–601,1993.

[11] Richard G. Heck, Jr.The Consistency of Predicative Fragments ofFrege’s Grundgesetze der Arithmetik.History and Philosophy of Logic, 17(4):209–220,1996.

[12] Stephen Cole Kleene.Quantification of Number-Theoretic Functions.Compositio Mathematica, 14:23–40, 1959.

[13] Øystein Linnebo.Predicative Fragments of Frege Arithmetic.The Bulletin of Symbolic Logic, 10(2):153–174,2004.

[14] Gerald E. Sacks.Higher Recursion Theory.


Perspectives in Mathematical Logic.Springer-Verlag, Berlin, 1990.

[15] Stephen G. Simpson.Subsystems of Second Order Arithmetic.Perspectives in Mathematical Logic.Springer-Verlag, Berlin, 1999.

[16] John R. Steel.Forcing with Tagged Trees.Annals of Mathematical Logic, 15(1):55–74,1978.

[17] Lou van den Dries.Tame Topology and O-Minimal Structures,volume 248 of London Mathematical SocietyLecture Note Series.Cambridge University Press, Cambridge, 1998.

[18] Albert Visser.The Predicative Frege Hierarchy.Annals of Pure and Applied Logic,160(2):129–153, 2009.

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