number systems of different lengths, and a natural ...tf/pettigrew2.pdf · ea systems examples...

EA Systems Examples Induction and Recursion Length Measuring the Universe Analysis

Number systems of different lengths,and a natural approach to infinitesimal analysis

Richard PettigrewDepartment of Mathematics

University of Bristol

April 21, 2008

[email protected] Natural number systems and infinitesimal analysis

Outline

The talk in brief

Describe EA, a finitary theory of finite sets.

Define the notion of a natural number system in EA.

Show that there are non-isomorphic natural number systemsin EA.

Give a taste of the theory of natural number systems in EA.

Sketch the beginnings of a theory of infinitesimal analysis inan extension on EA.

Outline

The talk in brief

Outline

The talk in brief

Outline

The talk in brief

Outline

The talk in brief

The language of EA

Constants, operators, and function and relation symbols

Constant: ∅ (empty set)

Function symbols:

P (power set)TC (transitive closure){ , } (pair set)

Term-forming operator: {x ∈ t : A(x)}, whenever A isbounded.

Relation symbols:

= (identity)∈ (membership)

The language of EA

Function symbols:

Relation symbols:

The language of EA

Function symbols:

Relation symbols:

The language of EA

Function symbols:

Relation symbols:

The axioms of EA

EA in a nutshell

Axioms of Empty Set, Power Set, Transitive Closure, Pair Set,and Extensionality are definitions of primitive symbols.

Instead of the Axiom of Infinity, EA has the Axiom ofDedekind Finiteness: ∀x , y(x $ y → x <c y)

The Axiom Schema of Subset Comprehension is restricted tobounded formulae.

In other words...

EA is finitely axiomatized by adding the Axioms of DedekindFiniteness, P, and TC to Jensen’s rudimentary functions.

EA is mutually interpretable with I∆0 + exp. (With mutuallyinverse interpretations? Don’t know. Suspect not.)

The axioms of EA

EA in a nutshell

In other words...

The axioms of EA

EA in a nutshell

In other words...

The axioms of EA

EA in a nutshell

In other words...

The axioms of EA

EA in a nutshell

In other words...

The axioms of EA

EA in a nutshell

In other words...

Definitions

Definition (L generated from 0 by σ)

Suppose σ is a unary global function defined by a term of EA, and0 is a closed term. Then, if L is a linear ordering, we say that L isgenerated from 0 by σ if

(1) First(L) = 0, and

(2) NextL(x) = σ(x), for all x in Field(L) except Last(L).

Thus, roughly, if L is generated from 0 by σ, it has the followingform:

[0, σ(0), σ(σ(0)), · · · , a]

Definitions

Definition (Natural number system)

We say that σ generates a natural number system from 0 if

∀L(L is generated from 0 by σ → σ(Last(L)) 6∈ Field(L))

If N = (σN , 0N ) is a natural number system, we say that L isin N if L is generated from 0N by σN .

If L = [0N , · · · a] in N , let σN (L) = [0N , · · · , a, σN (a)].

Thus, roughly, N consists of the following linear orderings:

[ ], [0N ], [0N , σN (0N )], [0N , σN (0N ), σN (σN (0N ))], · · ·

Definitions

[ ], [0N ], [0N , σN (0N )], [0N , σN (0N ), σN (σN (0N ))], · · ·

Definitions

[ ], [0N ], [0N , σN (0N )], [0N , σN (0N ), σN (σN (0N ))], · · ·

Definitions

[ ], [0N ], [0N , σN (0N )], [0N , σN (0N ), σN (σN (0N ))], · · ·

Examples of natural number systems

Example

VN is generated from ∅ by σVN : x 7→ x ∪ {x}

[ ], [∅], [∅, {∅}], [∅, {∅}, {∅, {∅}}], · · ·

Z is generated from ∅ by σZ : x 7→ {x}

[ ], [∅], [∅, {∅}], [∅, {∅}, {{∅}}], · · ·

CH is generated from ∅ by σCH : x 7→ P(x)

[ ], [∅], [∅,P(∅)], [∅,P(∅),P(P(∅))], · · ·

Example

[ ], [∅], [∅, {∅}], [∅, {∅}, {∅, {∅}}], · · ·

[ ], [∅], [∅, {∅}], [∅, {∅}, {{∅}}], · · ·

[ ], [∅], [∅,P(∅)], [∅,P(∅),P(P(∅))], · · ·

Example

[ ], [∅], [∅, {∅}], [∅, {∅}, {∅, {∅}}], · · ·

[ ], [∅], [∅, {∅}], [∅, {∅}, {{∅}}], · · ·

[ ], [∅], [∅,P(∅)], [∅,P(∅),P(P(∅))], · · ·

Example

[ ], [∅], [∅, {∅}], [∅, {∅}, {∅, {∅}}], · · ·

[ ], [∅], [∅, {∅}], [∅, {∅}, {{∅}}], · · ·

[ ], [∅], [∅,P(∅)], [∅,P(∅),P(P(∅))], · · ·

Induction along natural number systems

Theorem (Bounded induction)

Suppose A is a bounded formula of EA. Then

EA ` (A( [ ] ) & (∀L in N )[A(L) → A(σN (L))]) → (∀L in N )A(L)

Unbounded induction DOES NOT HOLD

If A is unbounded, then the following does not necessarily hold:

Both are consequences of the presence only of bounded separation.

Recursion along a natural number system

Definition (Arithmetical global functions)

Suppose ϕ is a global function. We say that ϕ is arithmetical if

EA ` ∀x , y(x ∼= y → ϕ(x) ∼= ϕ(y))

Definition (N is closed under ϕ)

Suppose ϕ is an arithmetical global function. Then we say that Nis closed under ϕ if

EA ` (∀x in N )(∃y in N )[Field(y) ∼= ϕ(Field(x))].

Definition (Arithmetical global functions)

Suppose ϕ is a global function. We say that ϕ is arithmetical if

EA ` ∀x , y(x ∼= y → ϕ(x) ∼= ϕ(y))

Definition (N is closed under ϕ)

Suppose ϕ is an arithmetical global function. Then we say that Nis closed under ϕ if

EA ` (∀x in N )(∃y in N )[Field(y) ∼= ϕ(Field(x))].

Theorem

Given a natural number system N , the family of arithmetical globalfunctions under which N is closed is closed under limited recursion.

Unlimited recursion DOES NOT HOLD

Given a natural number system N , the family of arithmeticalglobal functions under which N is closed is NOT closed under fullrecursion.

Both are consequences of the presence only of bounded induction.Unlimited recursion requires Σ1 induction.

Theorem

For n = 0, 1, 2, 3, there are natural number systems closedunder all and only the arithmetical functions of Grzegorczyk’sclass E n:

VN , Z, CH closed under ϕ ⇔ ϕ ∈ E 0.There is a system, ACK, which moves from one set to anotherin their Ackermann ordering:

ACK is closed under ϕ ⇔ ϕ ∈ E 3.

i.e ACK closed under exponentiation, but notsuperexponentiation.

But the distinctions are more fine-grained:

There are natural number systems closed under x + log(x) butnot under x + x .There are natural number systems closed under x log(log(x))but not under x log(x).

VN , Z, CH closed under ϕ ⇔ ϕ ∈ E 0.

There is a system, ACK, which moves from one set to anotherin their Ackermann ordering:

There are natural number systems closed under x + log(x) butnot under x + x .

There are natural number systems closed under x log(log(x))but not under x log(x).

Made-to-Measure Natural Number Systems

Definition (ϕ is maximally powerful in N )

ϕ is maximally powerful in N if, for any arithmetical globalfunction ψ, if N is closed under ψ, then there is n such that ψ iseventually majorized by ϕn.

Theorem

Suppose there is C such that

(i) EA ` (∀x)(C ≤ x → x < ϕ(x))

(ii) EA ` (∀x , y)(C ≤ x ≤ y → ϕ(x) ≤ ϕ(y))

(iIi) EA ` (∀x , y)(C ≤ x ≤ y → ϕ(x)− x ≤ 2y − y)

Then there a natural number system ACKϕ such that ϕ ismaximally powerful in ACKϕ.

Made-to-Measure Natural Number Systems

Definition (ϕ is maximally powerful in N )

ϕ is maximally powerful in N if, for any arithmetical globalfunction ψ, if N is closed under ψ, then there is n such that ψ iseventually majorized by ϕn.

Theorem

Suppose there is C such that

(i) EA ` (∀x)(C ≤ x → x < ϕ(x))

(ii) EA ` (∀x , y)(C ≤ x ≤ y → ϕ(x) ≤ ϕ(y))

(iIi) EA ` (∀x , y)(C ≤ x ≤ y → ϕ(x)− x ≤ 2y − y)

Then there a natural number system ACKϕ such that ϕ ismaximally powerful in ACKϕ.

Relations of length between natural numbers systems

Definition

M� N if

EA ` (∀x in M)(∃y in N )[Field(y) ∼= Field(x)].

i.e. there is an injection from M into N that preserves length.

Theorem

In the presence of Σ1 induction, and thus unlimited recursion, allnatural number systems are of the same length: i.e. M∼= N , forall M and N .

Relations of length between natural numbers systems

Definition

M� N if

EA ` (∀x in M)(∃y in N )[Field(y) ∼= Field(x)].

i.e. there is an injection from M into N that preserves length.

Theorem

In the presence of Σ1 induction, and thus unlimited recursion, allnatural number systems are of the same length: i.e. M∼= N , forall M and N .

The incommensurability of VN and Z

Theorem

VN and Z are incommensurable: that is,

VN 6� Z and Z 6� VN

I will sketch two proofs:

One is syntactic.

The other is model-theoretic.

The incommensurability of VN and Z

Theorem

VN and Z are incommensurable: that is,

VN 6� Z and Z 6� VN

I will sketch two proofs:

One is syntactic.

The other is model-theoretic.

The syntactic proof

Lemma (Parikh-style Bounding Lemma)

Suppose A is a bounded formula of EA. Then, if

EA ` ∀x∃!yA(x , y)

Then there is a classical natural number, n, such that

EA ` ∀x∃!y(y ∈ Pn(TC(x)) & A(x , y))

The syntactic proof

Proof. Suppose VN � Z. That is,

EA ` (∀v in VN )(∃!z in Z)(Field(v) ∼= Field(z))

Thus, by Parikh-style Bounding Lemma, there is n such that

EA ` (∀v in VN )(∃z! in Z)(z ∈ Pn(TC(v)) & Field(v) ∼= Field(z))

The syntactic proof

But, by (meta-theoretical) induction on n,

EA ` (∀v in VN )(∀z in Z)(z ∈ Pn(TC(v)) → z ∈ Vn+4)

EA ` (∀v in VN )(∃z! in Z)(z ∈ Vn+4 & Field(v) ∼= Field(z))

which is false. 2

The model-theoretic proof

Proof. Let M be a model of EA that contains a non-standardmember of VN , b. Then define the following submodel of M:

C (M, b) =∞⋃

{x ∈ M : M |= x ∈ Pn(b)}

We call C (M, b) the cumulation model of EA of b. Then

C (M, b) |= EA

But C (M, b) contains only standard members of Z, while itcontains non-standard members of VN . Thus, it is not the casethat VN � Z. 2

Measuring the universe

Definition (N measures the universe)

N measures the universe if

EA ` (∀x)(∃y in N )[x ∼= Field(y)]

Theorem

In the presence of Σ1 induction, and thus unlimited recursion,every natural number system measures the universe.

Theorem

In EA, no natural number system measures the universe.

Proof. Suppose N measures the universe. If k is a classicalnatural number, let

vk be the kth member of VN ,

zk be the kth member of Z, and

nk be the kth member of N .

Since N measures the universe,

EA ` (∀x)(∃y ! in N )[x ∼= y ]

Thus, by the Parikh-style Bounding Lemma, there is n such that

EA ` (∀x)(∃y ! in N )[y ∈ Pn(x) & x ∼= y ]

Thus, for all classical natural numbers, k,

nk ∈ Pn(vk) and nk ∈ Pn(zk)

Thus,nk ∈ Pn(vk) ∩ Pn(zk)

Thus,nk ∈ Vn+4

But this gives a contradiction, since Vn+4 cannot containsufficiently many members of N to measure all standard membersof VN and Z. 2

Extending EA

Definition (N -small and N -large)

Suppose N is a natural number system.

x is N -small ↔ (∃y in N )[x < Field(y)]

x is N -large ↔ (∀y in N )[Field(y) < x ]

Definition

EA+ is obtained from EA by adding the following axiom:

(∃x)[x is ACK-large]

Theorem

If EA is consistent, then EA+ is consistent.

Extending EA

Definition

Theorem

Extending EA

Definition

Theorem

Extending EA

Definition

Theorem

Extending EA

Definition

Theorem

Infinitesimal analysis in EA+

Definition (Integers in EA+)

An integer is an ordered pair (a, b) where a and b are sets.(Intuitively, (a, b) is a− b.)

(a, b) =Z (c , d) ↔ a + d ∼= b + c

Definition (Rationals in EA+)

A rational is an ordered pair (a, b) where a and b are integers, andb 6=Z 0. (Intuitively, (a, b) is a

(a, b) =Q (c , d) ↔ a×Z d ∼= b ×Z c

Definition (Integers in EA+)

An integer is an ordered pair (a, b) where a and b are sets.(Intuitively, (a, b) is a− b.)

(a, b) =Z (c , d) ↔ a + d ∼= b + c

Definition (Rationals in EA+)

A rational is an ordered pair (a, b) where a and b are integers, andb 6=Z 0. (Intuitively, (a, b) is a

(a, b) =Q (c , d) ↔ a×Z d ∼= b ×Z c

Definition (Reals in EA+)

r in R ↔ (∃x)[x is ACK-small & |r | < x ]

Definition (Infinitesimal in EA+)

r in I ↔ (∀x)

[x is ACK-small → |r | < 1

]Since there is an ACK-large set in EA+, there are infinitesimals.

Definition (Reals in EA+)

r in R ↔ (∃x)[x is ACK-small & |r | < x ]

Definition (Infinitesimal in EA+)

r in I ↔ (∀x)

[x is ACK-small → |r | < 1

]Since there is an ACK-large set in EA+, there are infinitesimals.

R is ‘almost’ real closed

Definition (x ' y)

If x and y are in R, then x ' y ↔ x − y in I

Theorem

If 0 < a is in R, then there is b in R such that b2 ' a.

If n is small and odd and {ai}ni=0 is a sequence of reals, thenthere is b in R such that

n∑i=0

aibi ' 0

R is ‘almost’ real closed

Definition (x ' y)

If x and y are in R, then x ' y ↔ x − y in I

Theorem

If 0 < a is in R, then there is b in R such that b2 ' a.

If n is small and odd and {ai}ni=0 is a sequence of reals, thenthere is b in R such that

n∑i=0

aibi ' 0

Continuous functions in EA+

Definition (f is continuous)

If f : J → R, then f is continuous if

(∀x , y in J)[x ' y → f (x) ' f (y)]

The following theorems hold:

The Intermediate Value Theorem

Every continuous function on a closed interval is bounded andattains its bounds.

(∀x , y in J)[x ' y → f (x) ' f (y)]

Differential and integral calculus in EA+

Definition (f is differentiable)

Suppose f : J → R, x is in J, and α is in R. Then f isdifferentiable at x with derivative α if

(∀δ in I )

[f (x + δ)− f (x)

δ' α

Definition (f is integrable)

Suppose f : [a, b] → R, a ≤ x ≤ b, and α is in R. Then f isintegrable at x with definite integral α if, for any ACK-large N,

N∑i=0

b − a

(a + i

b − a

Definition (f is differentiable)

Suppose f : J → R, x is in J, and α is in R. Then f isdifferentiable at x with derivative α if

(∀δ in I )

[f (x + δ)− f (x)

δ' α

Definition (f is integrable)

Suppose f : [a, b] → R, a ≤ x ≤ b, and α is in R. Then f isintegrable at x with definite integral α if, for any ACK-large N,

N∑i=0

b − a

(a + i

b − a

Rolle’s Theorem and the Mean Value Theorem.

The Fundamental Theorems of the Calculus

Polynomials of large degree

Definition

By definition,

N∑i=0

Theorem

For any large N and any x in R, exN is in R.

For any large M and N and any x in R, exM ' ex

Theorem

For any large N, the function x 7→ exN is differentiable at all points

x in R with derivative exN .

Definition

By definition,

N∑i=0

Theorem

Definition

By definition,

N∑i=0

Theorem

Weierstrass’ Approximation Theorem

Theorem (Weierstrass)

Suppose f : [a, b] → R is continuous function. Then there is apolynomial,

P(x) =N∑

possibly of large degree, such that

(∀a ≤ x ≤ b)[P(x) ' f (x)]

References

All the results here and many more can be found in:

Pettigrew, R. (doctoral thesis)Natural, Rational, and Real Arithmetic in the FinitaryTheory of Finite Setshttp://www.maths.bris.ac.uk/˜rp3959/thesis1.pdf/

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