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 Walsh
Department of Philosophy, Birkbeck, University of London,
and the Plurals, Predicates, and Paradox Project (ERC)
swalsh108@gmail.com or s.walsh@bbk.ac.uk
http://www.swalsh108.org
Logic Seminar, Göteborgs Universitet, 15 April 2011
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
Overview
Over the past 20 years, philosophers have studied the systems
closely related to subsystems of second-order arithmetic.
These systems are different in that they don’t have the ring
language but have some additional function symbols which take
the second-order part to the first-order part.
In this talk, we describe some new results on the interpretability
strength of these theories. In essence these results come down to
the 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 the
nature and limits of logicism in the philosophy of mathematics.
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
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 .
Subsystems of HP2
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
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, +, ×, ≤, ∈)
Subsystems of HP2
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
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.
Subsystems of HP2
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
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.
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
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 ∈.
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
An ω-sorted Language
More generally, we can consider a language for structures
(M, S1 , S2 , . . . , ∈1 , ∈2 , . . .)
where Sn ⊆
P(M n )
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 .
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
An ω-sorted Language
More generally, we can consider a language for structures
(M, S1 , S2 , . . . , ∈1 , ∈2 , . . .)
where Sn ⊆
P(M n )
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(M 2 ), . . .)
where M is any set and
(M, D1 (M), D2 (M), . . .)
where Dn (M) ⊆
Mn
are the definable subsets of structure M
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
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.
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
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
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
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)]
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
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 |
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
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)]
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
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)]
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
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)]
Russell’s paradox shows us that BL2 is inconsistent:
∃ X [∀ x x ∈ X ↔ (∃ Y ∂Y = x & x ∈
/ Y )]
Then
∂X ∈ X ↔ ∂X ∈
/X
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
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 Π11
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
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
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 Σ11 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).
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Provability Relation
The previous proposition tells us that the red theories prove the
black theories immediately below them.
Π11 − BL0
Σ11 − LB0
∆11 − BL0
ABL0
Π11 − CA0
Σ11 − AC0
∆11 − CA0
ACA0
Π11 − HP0
Σ11 − PH0
LLL
LLL
LLL
L%
rr
rrr
r
r
ry rr
∆11 − HP0
AHP0
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Provability Relation
The Russell paradox is derivable in Π11 − BL0 (since the set in
question is Σ11 -definable) so we remove it from the diagram.
Π11 − BL0
Σ11 − LB0
∆11 − BL0
ABL0
Π11 − CA0
Σ11 − AC0
∆11 − CA0
ACA0
Π11 − HP0
Σ11 − PH0
LLL
LLL
LLL
L%
rr
rrr
r
r
ry rr
∆11 − HP0
AHP0
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Provability Relation
The Russell paradox is derivable in Π11 − BL0 (since the set in
question is Σ11 -definable) so we remove it from the diagram.
Π11 − CA0
Σ11 − LB0
∆11 − BL0
ABL0
Σ11 − AC0
∆11 − CA0
ACA0
Π11 − HP0
Σ11 − PH0
LLL
LLL
LLL
L%
rr
rrr
r
r
ry rr
∆11 − HP0
AHP0
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Provability Relation
It is not obvious from the definitions, but one can in fact show that
Π11 − CA0 implies Σ11 − AC0 (see [15] Theorem V.8.3 p. 205).
Π11 − CA0
Σ11 − LB0
∆11 − BL0
ABL0
Σ11 − AC0
∆11 − CA0
ACA0
Π11 − HP0
Σ11 − PH0
LLL
LLL
LLL
L%
rr
rrr
r
r
ry rr
∆11 − HP0
AHP0
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Provability Relation
So here are the provability implications which we have now:
Π11 − CA0
Σ11 − LB0
∆11 − BL0
ABL0
Σ11 − AC0
∆11 − CA0
ACA0
Π11 − HP0
Σ11 − PH0
LLL
LLL
LLL
L%
rr
rrr
r
r
ry rr
∆11 − HP0
AHP0
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Provability Relation
In fact we can show the following, where ⇒ is irreversible
implication, → is reversal not known, and 9 is non-implication
Π11 − CA0
Σ11 − LB0
∆11 − BL0
ABL0
?
Σ11 − AC0
∆11 − CA0
ACA0
Π11 − HP0L m
LLLL
LLLL
LLLL
LLLL
L !)
|
- 1
Σ1 − PH0
rrrr
r
r
r
rr
rrrrr
u} rrrr
∆11 − HP0
AHP0
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
The Intepretability Relation
Another important relationship is that of interpretability:
Π11 − CA0 o
r
rrrr
rrrrr
r
r
r
r
u} rrrr
+
Σ11 − LB0 m ? Σ11 − AC0
Σ11 − LB0
∆11 − BL0
/ Π1 − HP
1
0
∆11 − CA0
ACA0 L
LLLLL
rr
LLLLLL
rrrrrr
r
r
r
r
LLLLLL
r
r
r
rrr
LL )!
r
r
r
t| r
? - 1
o
/
ABL0
Qj
Σ1 − PH0
Subsystems of HP2
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Intepretability Relation
The relation of interpretation between theories is a uniform version
of the relation of definability between structures.
Recall that M0 is interpretable in M (written M0 ≤I M) if some
isomorphic copy of M0 is definable in M.
Semantically, T 0 is interpretable in T (written T 0 ≤I T ) if every
model M of T uniformly interprets without parameters some
model M0 of T 0 , where uniformly means that the same formulas
are used each time.
Syntactically, T 0 is interpretable in T (written T 0 ≤I T ) if there is
a translation of the primitives of T 0 into formulas of T such that
the translations of T 0 -theorems are T -theorems.
E.g. PA2 ≤I ZFC or groups ≤I fields.
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Intepretability Relation
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 .
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Intepretability Relation
Frege’s project (cf. [10]) was to interpret PA2 within BL2 via HP2 :
BL2
/ HP2
/ PA2
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Intepretability Relation
Again, because its inconsistent, we remove BL2 from the diagram:
BL2
/ HP2
/ PA2
PA2 vs. BL2 vs. HP2
Subsystems of PA2
Philosophical Significance
Subsystems of BL2
Subsystems of HP2
The Intepretability Relation
Again, because its inconsistent, we remove BL2 from the diagram:
HP2
/ PA2
PA2 vs. BL2 vs. HP2
Subsystems of PA2
Philosophical Significance
The Intepretability Relation
Boolos [2] noted that PA2 interprets HP2 :
HP2
/ PA2
Subsystems of BL2
Subsystems of HP2
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
The Intepretability Relation
Boolos [2] noted that PA2 interprets HP2 :
HP2 o
/ PA2
Subsystems of BL2
Subsystems of HP2
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Intepretability Relation
Linnebo [13] noted this was uniform in the levels comprehension.
HP2 o
/ PA2
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Intepretability Relation
Linnebo [13] noted this was uniform in the levels comprehension.
Π1n − HP0 o
/ Π1 − CA
n
0
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Intepretability Relation
Linnebo [13] noted this was uniform in the levels comprehension.
Π11 − HP0 o
/ Π1 − CA
1
0
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Intepretability Relation
Linnebo [13] noted this was uniform in the levels comprehension.
Π11 − CA0 o
/ Π1 − HP
1
0
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
The Intepretability Relation
Since interpretability respects provability, we can add:
Π11 − CA0 o
/ Π1 − HP
1
0
Subsystems of HP2
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
The Intepretability Relation
Since interpretability respects provability, we can add:
Π11 − CA0 o
Σ11 − LB0
∆11 − BL0
ABL0
/ Π1 − HP
1
0
Σ11 − AC0
∆11 − CA0
ACA0
Σ11 − PH0
rr
rrr
r
r
r
y rr
∆11 − HP0
AHP0
Subsystems of HP2
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Intepretability Relation
Heck [11] showed ABL0 interprets Q and Burgess [4] did so for AHP0 :
Π11 − CA0 o
Σ11 − LB0
∆11 − BL0
ABL0
/ Π1 − HP
1
0
Σ11 − AC0
∆11 − CA0
ACA0
Σ11 − PH0
rr
rrr
r
r
r
y rr
∆11 − HP0
AHP0
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Intepretability Relation
Heck [11] showed ABL0 interprets Q and Burgess [4] did so for AHP0 :
Π11 − CA0 o
Σ11 − LB0
∆11 − BL0
ABL0 N
/ Π1 − HP
1
0
Σ11 − AC0
∆11 − CA0
Σ11 − PH0
NNN
pp
NNN
ppp
p
NNN
p
NN& xppppp
Q
∆11 − HP0
ACA0
rr
rrr
r
r
r
y rr
AHP0
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
The Intepretability Relation
Ganea [9] and Visser [18] showed that Q interprets ABL0 :
Π11 − CA0 o
Σ11 − LB0
∆11 − BL0
ABL0 N
/ Π1 − HP
1
0
Σ11 − AC0
∆11 − CA0
Σ11 − PH0
NNN
pp
NNN
ppp
p
NNN
p
NN& xppppp
Q
∆11 − HP0
ACA0
rr
rrr
r
r
r
y rr
AHP0
Subsystems of HP2
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
The Intepretability Relation
Ganea [9] and Visser [18] showed that Q interprets ABL0 :
Π11 − CA0 o
Σ11 − LB0
∆11 − BL0
/ Π1 − HP
1
Σ11 − AC0
∆11 − CA0
Σ11 − PH0
rr
rrr
r
r
r
y rr
∆11 − HP0
ACA0
ABL0 o
0
qq
qqq
q
q
q
x qqq
/Qq
AHP0
Subsystems of HP2
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Intepretability Relation
Ferreira & Wehmeier [6] show Π11 − CA0 implies Con(∆11 − BL0 ):
Π11 − CA0 o
Σ11 − LB0
∆11 − BL0
/ Π1 − HP
1
0
Σ11 − AC0
∆11 − CA0
Σ11 − PH0
ABL0 o
∆11 − HP0
ACA0
rr
rrr
r
r
r
y rr
qq
qqq
q
q
q
x qqq
/Qq
AHP0
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Intepretability Relation
Ferreira & Wehmeier [6] show Π11 − CA0 implies Con(∆11 − BL0 ):
Π11 − CA0 o
Σ11 − LB0 Σ11 − AC0
∆11 − BL0
∆11 − CA0
/ Π1 − HP
1
0
Σ11 − PH0
ABL0 o
∆11 − HP0
ACA0
rr
rrr
r
r
r
y rr
qq
qqq
q
q
q
x qqq
/Qq
AHP0
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Intepretability Relation
A slight modification shows Π11 − CA0 implies Con(Σ11 − LB0 ):
Π11 − CA0 o
Σ11 − LB0 Σ11 − AC0
∆11 − BL0
∆11 − CA0
/ Π1 − HP
1
0
Σ11 − PH0
ABL0 o
∆11 − HP0
ACA0
rr
rrr
r
r
r
y rr
qq
qqq
q
q
q
x qqq
/Qq
AHP0
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Intepretability Relation
A slight modification shows Π11 − CA0 implies Con(Σ11 − LB0 ):
Π11 − CA0 o
r
rrr
r
r
r
r
y rr
Σ11 − LB0
∆11 − BL0
/ Π1 − HP
1
0
Σ11 − AC0
∆11 − CA0
Σ11 − PH0
ABL0 o
∆11 − HP0
ACA0
rr
rrr
r
r
r
y rr
qq
qqq
q
q
q
x qqq
/Qq
AHP0
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
The Intepretability Relation
+
A new proof shows there is theory Σ11 − LB0 such that:
Π11 − CA0 o
r
rrr
r
r
r
r
y rr
-
+
Σ11 − LB0
Σ11 − LB0
∆11 − BL0
ABL0 o
/ Π1 − HP
1
0
Σ11 − AC0
∆11 − CA0
Σ11 − PH0
qq
qqq
q
q
qq
q
x qq
/Q
∆11 − HP0
ACA0
s
sss
s
s
s
y ss
s
AHP0
Subsystems of HP2
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
The Intepretability Relation
+
A new proof shows there is theory Σ11 − LB0 such that:
Π11 − CA0 o
r
rrr
r
r
r
r
y rr
-
+
Σ11 − LB0
Σ11 − LB0
∆11 − BL0
ABL0 o
/ Π1 − HP
1
0
Σ11 − AC0
∆11 − CA0
Σ11 − PH0
qq
qqq
q
q
qq
q
x qq
/Q
∆11 − HP0
ACA0
s
sss
s
s
s
y ss
s
AHP0
Subsystems of HP2
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
The Intepretability Relation
Another new proof shows that ACA0 proves Con(Σ11 − PH0 ):
Π11 − CA0 o
r
rrr
r
r
r
r
y rr
-
+
Σ11 − LB0
Σ11 − LB0
∆11 − BL0
ABL0 o
/ Π1 − HP
1
0
Σ11 − AC0
∆11 − CA0
Σ11 − PH0
qq
qqq
q
q
qq
q
x qq
/Q
∆11 − HP0
ACA0
s
sss
s
s
s
y ss
s
AHP0
Subsystems of HP2
PA2 vs. BL2 vs. HP2
Subsystems of PA2
Philosophical Significance
Subsystems of BL2
The Intepretability Relation
Another new proof shows that ACA0 proves Con(Σ11 − PH0 ):
Π11 − CA0 o
r
rrr
r
r
r
yrrr
- 1
+
Σ11 − LB0
Σ11 − LB0
Σ1 − AC0
∆11 − CA0
∆11 − BL0
ACA0 L
/Qo
ABL0 o
/ Π1 − HP
1
0
LLL
LLL
LLL
L%
Σ11 − PH0
Subsystems of HP2
PA2 vs. BL2 vs. HP2
Subsystems of PA2
Philosophical Significance
Subsystems of BL2
Subsystems of HP2
The Intepretability Relation
Since the arrows we get from proving consistency are irreversible:
Π11 − CA0 o
r
rrr
r
r
r
yrrr
- 1
+
Σ11 − LB0
Σ11 − LB0
Σ1 − AC0
∆11 − CA0
∆11 − BL0
ACA0 L
/Qo
ABL0 o
/ Π1 − HP
1
0
LLL
LLL
LLL
L%
Σ11 − PH0
PA2 vs. BL2 vs. HP2
Subsystems of PA2
Philosophical Significance
Subsystems of BL2
Subsystems of HP2
The Intepretability Relation
Since the arrows we get from proving consistency are irreversible:
Π11 − CA0 o
r
rrrr
rrrrr
r
r
r
r
u} rrrr
- 1
+
Σ11 − LB0
Σ11 − LB0
∆11 − BL0
/ Π1 − HP
1
0
Σ1 − AC0
∆11 − CA0
ACA0 L
LLLLL
rr
LLLLLL
rrrrrr
r
r
r
r
LLLLLL
r
r
r
rrr
LL )!
r
r
r
t| r
o
/
o
ABL0
Q
Σ11 − PH0
PA2 vs. BL2 vs. HP2
Subsystems of PA2
Philosophical Significance
Subsystems of BL2
The Intepretability Relation
Π11 − CA0 o
rrr
rrrrr
r
r
r
rrr
u} rrrr
- 1
+
Σ11 − LB0
Σ11 − LB0
∆11 − BL0
/ Π1 − HP
1
0
Σ1 − AC0
∆11 − CA0
ACA0 L
LLLLL
rr
LLLLLL
rrrrrr
r
r
r
LLLLLL
rrr
r
r
r
r
LL )!
r
r
r
t| r
/Qo
ABL0 o
Σ11 − PH0
Subsystems of HP2
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
The Intepretability Relation
Adding credits and the some of the open questions:
Frege/Boolos/Linnebo
/ Π1 − HP
Π11 − CA0 o
1
0
r
rrrr
rrrrr
r
r
rr
u} rrrrWalsh +
Σ11 − LB0 m ? Σ11 − AC0
Σ11 − LB0
∆11 − BL0
∆11 − CA0
ACA0 L
LLLLL
rr
LLLLLWalsh
rrrrrr
r
LLLL
r
r
r
r
r
LLLLL
r
rrr
r
r
r
)!
t| r
? - 1
/Qj
ABL0 o
Σ1 − PH0
Subsystems of HP2
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
Logicism about Arithmetical Knowledge
The logicist’ idea is that arithmetical knowledge is grounded in
logical knowledge, viz. Frege on mathematical induction:
“One will be able to see from this essay that even inferences which
are apparently particular to mathematics, like the inference from n
to n + 1, are based on general logical laws, so that they do not
require particular laws of aggregative thought” ([7] p. iv, cf. § 80
p. 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 our
making’ must recognise the question of what account should be
given of our ability to apprehend their truth. If Frege’s attempt to
ground that apprehension in pure logic were to succeed, we should
have an answer [. . . ] ” ([19] p. 131, cf. p. xiv).
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
Two Central Tasks of Logicism
The logicist account thus attempts to reduce our knowledge of
mathematical induction and the Peano axioms to our knowledge of
more 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 from
knowledge of these basic truths to knowledge of the Peano axioms.
. . . Given Frege’s result that PA2 is interpretable in HP2 , it is thus
natural to take the notion of reduction from (ii) to be that of
interpretability. This suggests the following version of the logicist
argument.
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
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 axioms
are interpretable in Hume’s Principle.
Preservation Premise: If it is known that principles P are
interpretable in principles P ∗ , and principles P ∗ are known,
then principles P are known.
Conclusion: The Peano axioms are known.
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
Technical Questions Motivated by The Logicist Template
I wanted to understand how flexible the interpretability premise is.
Interpretability Premise: It is known that the Peano axioms
are interpretable in Hume’s Principle.
In particular, I wanted to know:
1. Is this premise sensitive to the difference between predicative
and impredicative comprehension? Is there predicative version
of 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 subsystem
of Basic Law V?
This is still open. I conjecture: answer is “no.” Today I
focus on preliminary step of building models of Basic Law V
which are mutually interpretable with the natural numbers.
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
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 Π11 .
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
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
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 = ϕe a
I
Y ∈ LωCK [X ], where Lα [X ] is Gödel’s L up to α relative to X .
HX
1
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
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 ω1X be the least non-X -computable ordinal and let ω1CK be the
least non-computable ordinal.
If α is X -computable then α + 1 is X -computable, and
if αn is uniformly X -computable where αn < αn+1
then supn αn is X -computable.
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
Kleene’s Construction: Kleene’s O
Kleene’s OX is a Π1,X
1 -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 → ω1X
1
7→
0
a
7→
3 · 5e
7→
|a| + 1
sup ϕX
(n)
e
2
n
X
where ϕX
e (n) < ϕe (n + 1)
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Kleene’s Construction: The H-sets
The sets HaX for a ∈ OX are defined as follows:
H1X = X
H2Xa = (HaX )0
X
X
H3·5
e = ⊕n H X
ϕ (n)
e
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 = ϕe a
HX
Subsystems of HP2
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
Kleene’s Construction: The Key Theorem
Kleene [12] proved the following important theorem:
Theorem
The predicate ∃ Y ≤h X ϕ(Y , X ) where ϕ is Π11 is a Π11 -predicate.
There is a converse to this, called the Spector-Gandy theorem:
Theorem
If ψ(X ) is a Π11 -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.
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
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 Σ11 .
Hence, the set of X = {n : ∃ Y ≤h ∅ ϕ(Y , n)} is ∆11
and so by Kleene’s result X ∈ HYP.
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Barwise-Schlipf Construction
Barwise & 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 every
X -recursive partial type p(v ) over a finite number of parameters
is realized in the model.
Proposition
For every countable model M and every X ∈ 2ω , there is a
countable X -recursively saturated elementary extension N M
This proposition is in fact provable in WKL0 (cf. [15] p. 383).
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Barwise-Schlipf Construction
Suppose N is recursively saturated and θi is computable. Then:
[∀ K > 0 N |= ∃ n
K
^
¬θi (n)] =⇒ [N |= ∃ n
i=1
^
¬θi (n)]
i
Note that the contrapositive is the following:
[N |= ∀ n
_
i
θi (n)] =⇒ [∃ K > 0 N |= ∀ n
K
_
i=1
θi (n)]
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
The Barwise-Schlipf Construction
Theorem
Suppose 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 .
Theorem
Suppose that M |= ∀ n ∃ Y ϕ(Y , n) ↔ ¬∃ Y ψ(Y , n)
where ϕ and ψ are arithmetical. Then
N |= ∀ n
_
∃ b ϕ(θ(·, b), n) ∨ ¬ψ(θ(·, b), n)
θ
Then there are θ1 , . . . , θK such that
N |= ∀ n
K
_
i=1
∃ b ϕ(θi (·, b), n) ∨ ¬ψ(θi (·, b), n)
Subsystems of HP2
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
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
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
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 ∂ϕs : M n → M
where n is the number of parameter variables in ϕ, so that
Ms |= ∂ϕs (a) = ∂ψt (b) ↔ ∀ z ϕ(z, a) ↔ ψ(z, b)
Then we let Mω =
S
s
Ms in signature Lω =
S
s
Ls
PA2 vs. BL2 vs. HP2
Subsystems of PA2
Philosophical Significance
Subsystems of BL2
Subsystems of HP2
Ferreira & Wehmeier’s Barwise-Schlipf Model of Σ11 − LB0
The second step is to take an recursively saturated N Mω ,
and let M = (N, D1 (N ), D2 (N ), . . . , ∂) where ∂(θ(·, a)) = ∂θs (a).
Suppose that M |= ∀ n ∃ X ϕ(n, X , ∂(X )) where ϕ is arithmetical.
N |= ∀ n
_
∃ b ϕ(n, θ(·, b), ∂θs (b))
θ
Then there are θ1 , . . . , θK such that
N |= ∀ n
K
_
∃ b ϕ(n, θi (·, b), ∂θsi (b))
i=1
Since we have definable skolem functions, there is definable f
N |= ∀ n
K
_
i=1
ϕ(n, θi (·, f (n)), ∂θsi (f (n)))
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
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
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
A Kleene Model of Σ11 − LB0
As a first step let us consider the relation
a
Q(X , a, e) ≡ X ∈ HYP & a ∈ O & e ∈ ω & X = ϕH
e
After recalling some definitions, we see that Q is a Π11 -relation.
By Π11 -uniformization, there is Π11 -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 ) = ha, ei iff q(X , a, e)
Note by Spector-Gandy that graph(∂) is definable over (ω, HYP).
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
A Kleene Model of Σ11 − LB0
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 Σ11 .
Hence, this set is ∆11 and so by Kleene’s result is in HYP.
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
A Kleene Model of Σ11 − LB0
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
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
A Kleene Model of Σ11 − LB0
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 Π11 − CA0 .
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
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 .
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Boolos’ Interpretation in HP2 in PA2
Boolos [2] noted that HP2 is interpretable in PA2 :
Working in PA2 we can define the # function by
(
0
if X infinite,
#X =
|X | + 1 otherwise.
Subsystems of HP2
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Boolos’ Interpretation in HP2 in PA2
Boolos [2] noted that HP2 is interpretable in PA2 :
Working in PA2 we can define the # function by
(
0
if X infinite,
#X =
|X | + 1 otherwise.
This gives us the following information about provability:
?
Π11 − HP0L m
LLLL
LLLL
LLLL
LLLL
L !)
|
- 1
Σ1 − PH0
rr
r
r
rrrr
r
r
rrrr
u} rrrr
∆11 − HP0
AHP0
Subsystems of HP2
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of 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
(
0
if X infinite,
#X =
|X | + 1 otherwise.
And this gives us the following information about interpretability:
Σ11 − AC0
∆11 − CA0
ACA0
- 1
Σ1 − PH0
- 1 ∆1 − HP0
,
AHP0
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
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.
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
A Barwise-Schlipf Model of Σ11 − PH0
Suppose that X is a definable subset of R n .
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
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
A Barwise-Schlipf Model of Σ11 − PH0
For each m, we inductively define a decomposition of R m .
A decomposition of R 1 has the following form:
{(−∞, a1 ), (a1 , a2 ), . . . , (ak , +∞), {a1 }, . . . , {ak }}
where a1 < a2 < · · · < ak
A decomposition of R m+1 = R m × R is a finite partition of R m+1
into cells A such that the set of projections π(A) is a
decomposition of R m .
Theorem
(Cell Decomposition) For any finite sequence of B-definable sets
A1 , . . . , Ak ⊆ R m , there is a decomposition of R m partitioning
each of the Ai . Moreover, the cells in the decomposition are
B-definable.
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
A Barwise-Schlipf Model of Σ11 − PH0
The dimension of a definable set is defined as follows:
dim(X ) = max{i1 + · · · + in : X contains a (i1 , . . . , in )-cell}
For example X ⊆ R 1 , then dim(X ) > 0 iff X contains an interval.
Note that X ⊆ R n 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
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
A Barwise-Schlipf Model of Σ11 − PH0
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.
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
A Barwise-Schlipf Model of Σ11 − PH0
Theorem
Suppose that R is an o-minimal expansion of a real closed field.
Suppose that X ⊆ R n and Y ⊆ R m 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.
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
A Barwise-Schlipf Model of Σ11 − PH0
Theorem
Suppose that R is an o-minimal expansion of a real closed field.
Suppose that X ⊆ R n and Y ⊆ R m 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 = hdim(X ), E (X )i
Then by the theorem, M models Hume’s Principle.
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
A Barwise-Schlipf Model of Σ11 − PH0
Suppose M |= ∀ n ∃ Y ϕ(n, Y , #Y ) where ϕ is arithmetical. Then
R |= ∀ n
_
∃ b ϕ(n, θ(·, b), hdim(θ(·, b)), E (θ(·, b))i)
θ
Since dim(X ) & E (X ) are bounded & definable, these are formulas.
Then there are θ1 , . . . , θK such that
K
_
R |= ∀ n
∃ b ϕ(n, θi (·, b), hdim(θi (·, b)), E (θi (·, b))i)
i=1
Since R has definable skolem functions there is a definable f
R |= ∀ n
K
_
i=1
ϕ(n, θi (·, f (n)), hdim(θi (·, f (n))), E (θi (·, f (n)))i)
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
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 Π11 − CA0 .
I
Σ11 − PH0 <I ACA0 .
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
The Intepretability Relation
Π11 − CA0 o
rrr
rrrrr
r
r
r
rrr
u} rrrr
+
Σ11 − LB0 m ? Σ11 − AC0
Σ11 − LB0
∆11 − BL0
/ Π1 − HP
1
0
∆11 − CA0
ACA0 L
LLLLL
rr
LLLLLL
rrrrrr
r
r
r
LLLLLL
rrr
r
r
r
r
LL
r
r
r
t| r
- )! 1
?
/Qj
ABL0 o
Σ1 − PH0
Subsystems of HP2
PA2 vs. BL2 vs. HP2
Philosophical Significance
Subsystems of PA2
Subsystems of BL2
Subsystems of HP2
The Provability Relation
Π11 − CA0
Σ11 − LB0
∆11 − BL0
ABL0
?
Σ11 − AC0
∆11 − CA0
ACA0
Π11 − HP0L m
LLLL
LLLL
LLLL
LLLL
L !)
|
- 1
Σ1 − PH0
rr
r
r
rrrr
r
r
rrrr
u} rrrr
∆11 − HP0
AHP0
PA2 vs. BL2 vs. HP2
[1]
[2]
Philosophical Significance
Subsystems of PA2
Jon Barwise and John Schlipf.
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[3]
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[4]
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[5]
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[6]
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Subsystems of BL2
Subsystems of HP2
[7]
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[8]
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[9]
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2004.
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PA2 vs. BL2 vs. HP2
Philosophical Significance
Perspectives in Mathematical Logic.
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[15] Stephen G. Simpson.
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Subsystems of PA2
Subsystems of BL2
Subsystems of HP2