Reverse Mathematics & Nonstandard Analysis: Making sense of - - PowerPoint PPT Presentation

Reverse Mathematics & Nonstandard Analysis:

Making sense of infinite computations.

Sam Sanders1

Tohoku University & Ghent University

CiE, June 29, 2011, Sofia

1This research is generously supported by the John Templeton Foundation.

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Goal & Motivation

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Goal & Motivation

Goal: To formalize Computability using Nonstandard Analysis.

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Goal & Motivation

Goal: To formalize Computability using Nonstandard Analysis. Motivation:

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Goal & Motivation

Goal: To formalize Computability using Nonstandard Analysis. Motivation: Erret Bishop (and others) have derided NSA for its ‘lack of computational content’.

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Nonstandard Analysis

◆

◆ ◆

◆ ◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Nonstandard Analysis

Nonstandard Analysis formalizes ‘calculus with infinitesimals’. ◆

◆ ◆

◆ ◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Nonstandard Analysis

Nonstandard Analysis formalizes ‘calculus with infinitesimals’. For this talk, we only need ∗◆.

◆ ◆

◆ ◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Nonstandard Analysis

Nonstandard Analysis formalizes ‘calculus with infinitesimals’. For this talk, we only need ∗◆.

✲

1 2 3 . . .

◆ ◆

◆ ◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Nonstandard Analysis

Nonstandard Analysis formalizes ‘calculus with infinitesimals’. For this talk, we only need ∗◆.

✲

1 2 3 . . .

• ◆, the natural numbers

◆

◆ ◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Nonstandard Analysis

Nonstandard Analysis formalizes ‘calculus with infinitesimals’. For this talk, we only need ∗◆.

✲

1 2 3 . . .

• ◆, the natural numbers

new numbers, not in ◆

• ◆

◆ ◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Nonstandard Analysis

Nonstandard Analysis formalizes ‘calculus with infinitesimals’. For this talk, we only need ∗◆.

✲

1 2 3 . . .

• ◆, the natural numbers

new numbers, not in ◆

• ω

◆

◆ ◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Nonstandard Analysis

Nonstandard Analysis formalizes ‘calculus with infinitesimals’. For this talk, we only need ∗◆.

✲

1 2 3 . . .

• ◆, the natural numbers

new numbers, not in ◆

• ω

ω − 1

◆

◆ ◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Nonstandard Analysis

Nonstandard Analysis formalizes ‘calculus with infinitesimals’. For this talk, we only need ∗◆.

✲

1 2 3 . . .

• ◆, the natural numbers

new numbers, not in ◆

• ω

ω − 1 2ω

◆

◆ ◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Nonstandard Analysis

Nonstandard Analysis formalizes ‘calculus with infinitesimals’. For this talk, we only need ∗◆.

✲

1 2 3 . . .

• ◆, the natural numbers

new numbers, not in ◆

• ω

ω − 1 2ω ⌈√ω⌉

◆

◆ ◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Nonstandard Analysis

Nonstandard Analysis formalizes ‘calculus with infinitesimals’. For this talk, we only need ∗◆.

✲

1 2 3 . . .

• ◆, the natural numbers

ω ω − 1 2ω ⌈√ω⌉

• infinite numbers

◆

◆ ◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Nonstandard Analysis

Nonstandard Analysis formalizes ‘calculus with infinitesimals’. For this talk, we only need ∗◆.

✲

1 2 3 . . .

• ◆, the finite numbers

ω ω − 1 2ω ⌈√ω⌉

• infinite numbers

◆

◆ ◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Nonstandard Analysis

Nonstandard Analysis formalizes ‘calculus with infinitesimals’. For this talk, we only need ∗◆.

✲

1 2 3 . . .

• ◆, the finite numbers

ω ω − 1 2ω ⌈√ω⌉

• infinite numbers

∗◆, the hypernatural numbers

• ◆

◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Nonstandard Analysis

Nonstandard Analysis formalizes ‘calculus with infinitesimals’. For this talk, we only need ∗◆.

✲

1 2 3 . . .

• ◆, the finite numbers

ω ω − 1 2ω ⌈√ω⌉

• infinite numbers

∗◆, the hypernatural numbers

• ◆= finite (or standard) numbers

◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Nonstandard Analysis

Nonstandard Analysis formalizes ‘calculus with infinitesimals’. For this talk, we only need ∗◆.

✲

1 2 3 . . .

• ◆, the finite numbers

ω ω − 1 2ω ⌈√ω⌉

• infinite numbers

∗◆, the hypernatural numbers

• ◆= finite (or standard) numbers

∗◆ \ ◆= infinite (or nonstandard) numbers

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

A simple example

◆ ❆ ◆ ❆ ❇ ◆ ❇ ❇

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

A simple example

Suppose we have proved (∃n ∈ ◆)ϕ(n). (ϕ is quantifier-free) ❆ ◆ ❆ ❇ ◆ ❇ ❇

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

A simple example

Suppose we have proved (∃n ∈ ◆)ϕ(n). (ϕ is quantifier-free) Program ❆ to find n0 ∈ ◆ s.t. ϕ(n0): ❆ ❇ ◆ ❇ ❇

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

A simple example

Suppose we have proved (∃n ∈ ◆)ϕ(n). (ϕ is quantifier-free) Program ❆ to find n0 ∈ ◆ s.t. ϕ(n0): 0) Define m := 0. ❆ ❇ ◆ ❇ ❇

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

A simple example

Suppose we have proved (∃n ∈ ◆)ϕ(n). (ϕ is quantifier-free) Program ❆ to find n0 ∈ ◆ s.t. ϕ(n0): 0) Define m := 0. 1) Check ϕ(m). ❆ ❇ ◆ ❇ ❇

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

A simple example

Suppose we have proved (∃n ∈ ◆)ϕ(n). (ϕ is quantifier-free) Program ❆ to find n0 ∈ ◆ s.t. ϕ(n0): 0) Define m := 0. 1) Check ϕ(m). 2) If ϕ(m) is TRUE, return m, ❆ ❇ ◆ ❇ ❇

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

A simple example

Suppose we have proved (∃n ∈ ◆)ϕ(n). (ϕ is quantifier-free) Program ❆ to find n0 ∈ ◆ s.t. ϕ(n0): 0) Define m := 0. 1) Check ϕ(m). 2) If ϕ(m) is TRUE, return m, otherwise define m := m + 1 and go to 1). ❆ ❇ ◆ ❇ ❇

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

A simple example

Suppose we have proved (∃n ∈ ◆)ϕ(n). (ϕ is quantifier-free) Program ❆ to find n0 ∈ ◆ s.t. ϕ(n0): 0) Define m := 0. 1) Check ϕ(m). 2) If ϕ(m) is TRUE, return m, otherwise define m := m + 1 and go to 1). The program ❆ will certainly halt. ❇ ◆ ❇ ❇

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

A simple example

Suppose we have proved (∃n ∈ ◆)ϕ(n). (ϕ is quantifier-free) Program ❆ to find n0 ∈ ◆ s.t. ϕ(n0): 0) Define m := 0. 1) Check ϕ(m). 2) If ϕ(m) is TRUE, return m, otherwise define m := m + 1 and go to 1). The program ❆ will certainly halt. Nonstandard program ❇ to find n0 ∈ ◆ s.t. ϕ(n0): ❇ ❇

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

A simple example

Suppose we have proved (∃n ∈ ◆)ϕ(n). (ϕ is quantifier-free) Program ❆ to find n0 ∈ ◆ s.t. ϕ(n0): 0) Define m := 0. 1) Check ϕ(m). 2) If ϕ(m) is TRUE, return m, otherwise define m := m + 1 and go to 1). The program ❆ will certainly halt. Nonstandard program ❇ to find n0 ∈ ◆ s.t. ϕ(n0): 0) For i = 0.. . . . ω do ❇ ❇

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

A simple example

Suppose we have proved (∃n ∈ ◆)ϕ(n). (ϕ is quantifier-free) Program ❆ to find n0 ∈ ◆ s.t. ϕ(n0): 0) Define m := 0. 1) Check ϕ(m). 2) If ϕ(m) is TRUE, return m, otherwise define m := m + 1 and go to 1). The program ❆ will certainly halt. Nonstandard program ❇ to find n0 ∈ ◆ s.t. ϕ(n0): 0) For i = 0.. . . . ω do If ϕ(i) is TRUE, return i and halt. ❇ ❇

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

A simple example

Suppose we have proved (∃n ∈ ◆)ϕ(n). (ϕ is quantifier-free) Program ❆ to find n0 ∈ ◆ s.t. ϕ(n0): 0) Define m := 0. 1) Check ϕ(m). 2) If ϕ(m) is TRUE, return m, otherwise define m := m + 1 and go to 1). The program ❆ will certainly halt. Nonstandard program ❇ to find n0 ∈ ◆ s.t. ϕ(n0): 0) For i = 0.. . . . ω do If ϕ(i) is TRUE, return i and

• halt. Continue otherwise.

❇ ❇

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

A simple example

Suppose we have proved (∃n ∈ ◆)ϕ(n). (ϕ is quantifier-free) Program ❆ to find n0 ∈ ◆ s.t. ϕ(n0): 0) Define m := 0. 1) Check ϕ(m). 2) If ϕ(m) is TRUE, return m, otherwise define m := m + 1 and go to 1). The program ❆ will certainly halt. Nonstandard program ❇ to find n0 ∈ ◆ s.t. ϕ(n0): 0) For i = 0.. . . . ω do If ϕ(i) is TRUE, return i and

• halt. Continue otherwise.

The program ❇ will certainly halt at some finite stage. ❇

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

A simple example

Suppose we have proved (∃n ∈ ◆)ϕ(n). (ϕ is quantifier-free) Program ❆ to find n0 ∈ ◆ s.t. ϕ(n0): 0) Define m := 0. 1) Check ϕ(m). 2) If ϕ(m) is TRUE, return m, otherwise define m := m + 1 and go to 1). The program ❆ will certainly halt. Nonstandard program ❇ to find n0 ∈ ◆ s.t. ϕ(n0): 0) For i = 0.. . . . ω do If ϕ(i) is TRUE, return i and

• halt. Continue otherwise.

The program ❇ will certainly halt at some finite stage. ❇ depends on the infinite ω, but not on the choice of ω.

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Turing Computability

We always assume that A ⊂ ◆ ⊂ ∗◆ and that ω is infinite. ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Turing Computability

We always assume that A ⊂ ◆ ⊂ ∗◆ and that ω is infinite.

Definition

The set A is ω-invariant if there is ψ ∈ ∆0 s.t. for all infinite ω , ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Turing Computability

We always assume that A ⊂ ◆ ⊂ ∗◆ and that ω is infinite.

Definition

The set A is ω-invariant if there is ψ ∈ ∆0 s.t. for all infinite ω , A = {k ∈ ◆ : ψ(k, ω)}.

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Turing Computability

We always assume that A ⊂ ◆ ⊂ ∗◆ and that ω is infinite.

Definition

The set A is ω-invariant if there is ψ ∈ ∆0 s.t. for all infinite ω , A = {k ∈ ◆ : ψ(k, ω)}. The set A depends on ω, but not on the choice of ω.

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Turing Computability

We always assume that A ⊂ ◆ ⊂ ∗◆ and that ω is infinite.

Definition

The set A is ω-invariant if there is ψ ∈ ∆0 s.t. for all infinite ω , A = {k ∈ ◆ : ψ(k, ω)}. The set A depends on ω, but not on the choice of ω.

Theorem

The ∆1-sets (=Turing computable) are exactly the ω-invariant sets.

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The Limit Lemma

◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The Limit Lemma

Theorem (Limit lemma)

f ≤T 0′ ⇐ ⇒ f ∈ ∆2 ⇐ ⇒ f = limn→∞ fn

(fn is computable)

◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The Limit Lemma

Theorem (Limit lemma)

f ≤T 0′ ⇐ ⇒ f ∈ ∆2 ⇐ ⇒ f = limn→∞ fn

(fn is computable)

0′ is a decision procedure for Σ1-formulas called ‘halting problem’. ◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The Limit Lemma

Theorem (Limit lemma)

f ≤T 0′ ⇐ ⇒ f ∈ ∆2 ⇐ ⇒ f = limn→∞ fn

(fn is computable)

0′ is a decision procedure for Σ1-formulas called ‘halting problem’.

Theorem (Hyperlimit Lemma)

f ≤T Π1 ⇐ ⇒ f ∈ ∆2 ⇐ ⇒ f = fω

(fn is computable)

◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The Limit Lemma

Theorem (Limit lemma)

f ≤T 0′ ⇐ ⇒ f ∈ ∆2 ⇐ ⇒ f = limn→∞ fn

(fn is computable)

0′ is a decision procedure for Σ1-formulas called ‘halting problem’.

Theorem (Hyperlimit Lemma)

f ≤T Π1 ⇐ ⇒ f ∈ ∆2 ⇐ ⇒ f = fω

(fn is computable)

Π1 is a decision procedure for Σ1-formulas, given by: ◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The Limit Lemma

Theorem (Limit lemma)

f ≤T 0′ ⇐ ⇒ f ∈ ∆2 ⇐ ⇒ f = limn→∞ fn

(fn is computable)

0′ is a decision procedure for Σ1-formulas called ‘halting problem’.

Theorem (Hyperlimit Lemma)

f ≤T Π1 ⇐ ⇒ f ∈ ∆2 ⇐ ⇒ f = fω

(fn is computable)

Π1 is a decision procedure for Σ1-formulas, given by:

Theorem (Π1)

For every ϕ ∈ ∆0, we have (∀n ∈ ◆)ϕ(n) → (∀n ∈ ∗◆)ϕ(n).

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The Limit Lemma

Theorem (Limit lemma)

f ≤T 0′ ⇐ ⇒ f ∈ ∆2 ⇐ ⇒ f = limn→∞ fn

(fn is computable)

0′ is a decision procedure for Σ1-formulas called ‘halting problem’.

Theorem (Hyperlimit Lemma)

f ≤T Π1 ⇐ ⇒ f ∈ ∆2 ⇐ ⇒ f = fω

(fn is computable)

Π1 is a decision procedure for Σ1-formulas, given by:

Theorem (Π1)

For every ϕ ∈ ∆0, we have (∀n ∈ ◆)ϕ(n) → (∀n ∈ ∗◆)ϕ(n). Also called ‘Transfer principle for Π1-formulas’ or ‘Π1-transfer’.

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Origins in RM

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Origins in RM

ERNA = Nonstandard Analysis in I∆0 + exp.

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Origins in RM

ERNA = Nonstandard Analysis in I∆0 + exp. RCA0 defines exactly the ∆1-sets.

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Origins in RM

ERNA = Nonstandard Analysis in I∆0 + exp. RCA0 defines exactly the ∆1-sets.

“Theorem”

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Origins in RM

ERNA = Nonstandard Analysis in I∆0 + exp. RCA0 defines exactly the ∆1-sets.

“Theorem”

• 1. If RCA0 proves T(=), then ERNA proves T(≈).

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Origins in RM

ERNA = Nonstandard Analysis in I∆0 + exp. RCA0 defines exactly the ∆1-sets.

“Theorem”

• 1. If RCA0 proves T(=), then ERNA proves T(≈).
• 2. If RCA0 proves [T(=) ⇔ WKL0],

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Origins in RM

ERNA = Nonstandard Analysis in I∆0 + exp. RCA0 defines exactly the ∆1-sets.

“Theorem”

• 1. If RCA0 proves T(=), then ERNA proves T(≈).
• 2. If RCA0 proves [T(=) ⇔ WKL0], then ERNA proves [T(≈) ⇔ Π1-TRANS].

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Origins in RM

ERNA = Nonstandard Analysis in I∆0 + exp. RCA0 defines exactly the ∆1-sets.

“Theorem”

• 1. If RCA0 proves T(=), then ERNA proves T(≈).
• 2. If RCA0 proves [T(=) ⇔ WKL0], then ERNA proves [T(≈) ⇔ Π1-TRANS].

Here, T(=) is a theorem of ordinary Mathematics.

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Origins in RM

ERNA = Nonstandard Analysis in I∆0 + exp. RCA0 defines exactly the ∆1-sets.

“Theorem”

• 1. If RCA0 proves T(=), then ERNA proves T(≈).
• 2. If RCA0 proves [T(=) ⇔ WKL0], then ERNA proves [T(≈) ⇔ Π1-TRANS].

Here, T(=) is a theorem of ordinary Mathematics. Example of 1: Intermediate Value Theorem.

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Origins in RM

ERNA = Nonstandard Analysis in I∆0 + exp. RCA0 defines exactly the ∆1-sets.

“Theorem”

• 1. If RCA0 proves T(=), then ERNA proves T(≈).
• 2. If RCA0 proves [T(=) ⇔ WKL0], then ERNA proves [T(≈) ⇔ Π1-TRANS].

Here, T(=) is a theorem of ordinary Mathematics. Example of 1: Intermediate Value Theorem. Example of 2: Peano’s theorem for diff. eq. y′ = f (x, y).

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Origins in RM

ERNA = Nonstandard Analysis in I∆0 + exp. RCA0 defines exactly the ∆1-sets.

“Theorem”

• 1. If RCA0 proves T(=), then ERNA proves T(≈).
• 2. If RCA0 proves [T(=) ⇔ WKL0], then ERNA proves [T(≈) ⇔ Π1-TRANS].

Here, T(=) is a theorem of ordinary Mathematics. Example of 1: Intermediate Value Theorem. Example of 2: Peano’s theorem for diff. eq. y′ = f (x, y). But RCA0 and WKL0 are recursion theoretic!

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Origins in RM

ERNA = Nonstandard Analysis in I∆0 + exp. RCA0 defines exactly the ∆1-sets.

“Theorem”

• 1. If RCA0 proves T(=), then ERNA proves T(≈).
• 2. If RCA0 proves [T(=) ⇔ WKL0], then ERNA proves [T(≈) ⇔ Π1-TRANS].

Here, T(=) is a theorem of ordinary Mathematics. Example of 1: Intermediate Value Theorem. Example of 2: Peano’s theorem for diff. eq. y′ = f (x, y). But RCA0 and WKL0 are recursion theoretic! How about ERNA and Π1-TRANS?

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Constructive Reverse Mathematics

CRM = RM in Bishop’s ‘constructive analysis’. ◆ ◆ ◆ ◆ ◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Constructive Reverse Mathematics

CRM = RM in Bishop’s ‘constructive analysis’. An important principle is:

Principle (Σ1-excluded middle or LPO)

For every q.f. formula ϕ, we have (∃n ∈ ◆)ϕ(n) ∨ (∀n ∈ ◆)¬ϕ(n). ◆ ◆ ◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Constructive Reverse Mathematics

CRM = RM in Bishop’s ‘constructive analysis’. An important principle is:

Principle (Σ1-excluded middle or LPO)

For every q.f. formula ϕ, we have (∃n ∈ ◆)ϕ(n) ∨ (∀n ∈ ◆)¬ϕ(n). The previous principle states: There is a finite procedure that decides whether (∃n ∈ ◆)ϕ(n) or not. ◆ ◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Constructive Reverse Mathematics

CRM = RM in Bishop’s ‘constructive analysis’. An important principle is:

Principle (Σ1-excluded middle or LPO)

For every q.f. formula ϕ, we have (∃n ∈ ◆)ϕ(n) ∨ (∀n ∈ ◆)¬ϕ(n). The previous principle states: There is a finite procedure that decides whether (∃n ∈ ◆)ϕ(n) or not.

Principle (Π1-Transfer)

For every q.f. formula ϕ, we have (∃n ∈ ◆)ϕ(n) ∨ (∀n ∈ ∗◆)¬ϕ(n). ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Constructive Reverse Mathematics

CRM = RM in Bishop’s ‘constructive analysis’. An important principle is:

Principle (Σ1-excluded middle or LPO)

For every q.f. formula ϕ, we have (∃n ∈ ◆)ϕ(n) ∨ (∀n ∈ ◆)¬ϕ(n). The previous principle states: There is a finite procedure that decides whether (∃n ∈ ◆)ϕ(n) or not.

Principle (Π1-Transfer)

For every q.f. formula ϕ, we have (∃n ∈ ◆)ϕ(n) ∨ (∀n ∈ ∗◆)¬ϕ(n). The previous principle is equivalent to: There is an ω-invariant procedure that decides whether (∃n ∈ ◆)ϕ(n) or not.

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Constructive Reverse Mathematics

❘ ❘ ❘

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Constructive Reverse Mathematics

In CRM, LPO is equivalent to MCT and to ❘ ❘ ❘

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Constructive Reverse Mathematics

In CRM, LPO is equivalent to MCT and to

Principle

(∀x ∈ ❘)(x > 0 ∨ ¬(x > 0)) ❘ ❘

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Constructive Reverse Mathematics

In CRM, LPO is equivalent to MCT and to

Principle

(∀x ∈ ❘)(x > 0 ∨ ¬(x > 0)) The previous principle should be read: For x ∈ ❘, there is a finite procedure that decides if x > 0. ❘

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Constructive Reverse Mathematics

In CRM, LPO is equivalent to MCT and to

Principle

(∀x ∈ ❘)(x > 0 ∨ ¬(x > 0)) The previous principle should be read: For x ∈ ❘, there is a finite procedure that decides if x > 0. In NSA, Π1-TRANS is equivalent to MCT(≈) and to ❘

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Constructive Reverse Mathematics

In CRM, LPO is equivalent to MCT and to

Principle

(∀x ∈ ❘)(x > 0 ∨ ¬(x > 0)) The previous principle should be read: For x ∈ ❘, there is a finite procedure that decides if x > 0. In NSA, Π1-TRANS is equivalent to MCT(≈) and to

Principle

For x ∈ ❘, there is an ω-invariant procedure that decides if x > 0.

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Lost in translation

◆ ◆ ◆ ◆ ◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Lost in translation

Hypernegation provides a translation between NSA and CRM: ◆ ◆ ◆ ◆ ◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Lost in translation

Hypernegation provides a translation between NSA and CRM:

Definition (Hypernegation)

∼ [(∃n ∈ ◆)ϕ(n)] ◆ ◆ ◆ ◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Lost in translation

Hypernegation provides a translation between NSA and CRM:

Definition (Hypernegation)

∼ [(∃n ∈ ◆)ϕ(n)] ≡ (∀n ∈ ∗◆)¬ϕ(n). ◆ ◆ ◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Lost in translation

Hypernegation provides a translation between NSA and CRM:

Definition (Hypernegation)

∼ [(∃n ∈ ◆)ϕ(n)] ≡ (∀n ∈ ∗◆)¬ϕ(n). ∼ [(∀n ∈ ◆)ϕ(n)] ≡ (∃n ∈ ∗◆)¬ϕ(n). ◆ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Lost in translation

Hypernegation provides a translation between NSA and CRM:

Definition (Hypernegation)

∼ [(∃n ∈ ◆)ϕ(n)] ≡ (∀n ∈ ∗◆)¬ϕ(n). ∼ [(∀n ∈ ◆)ϕ(n)] ≡ (∃n ∈ ∗◆)¬ϕ(n). ∼ [(∀n ∈ ∗◆)ϕ(n)] ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Lost in translation

Hypernegation provides a translation between NSA and CRM:

Definition (Hypernegation)

∼ [(∃n ∈ ◆)ϕ(n)] ≡ (∀n ∈ ∗◆)¬ϕ(n). ∼ [(∀n ∈ ◆)ϕ(n)] ≡ (∃n ∈ ∗◆)¬ϕ(n). ∼ [(∀n ∈ ∗◆)ϕ(n)] ≡ (∃n ≤ ω)¬ϕ(n). ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Lost in translation

Hypernegation provides a translation between NSA and CRM:

Definition (Hypernegation)

∼ [(∃n ∈ ◆)ϕ(n)] ≡ (∀n ∈ ∗◆)¬ϕ(n). ∼ [(∀n ∈ ◆)ϕ(n)] ≡ (∃n ∈ ∗◆)¬ϕ(n). ∼ [(∀n ∈ ∗◆)ϕ(n)] ≡ (∃n ≤ ω)¬ϕ(n).

(ω is independent of parameters in ϕ)

◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Lost in translation

Hypernegation provides a translation between NSA and CRM:

Definition (Hypernegation)

∼ [(∃n ∈ ◆)ϕ(n)] ≡ (∀n ∈ ∗◆)¬ϕ(n). ∼ [(∀n ∈ ◆)ϕ(n)] ≡ (∃n ∈ ∗◆)¬ϕ(n). ∼ [(∀n ∈ ∗◆)ϕ(n)] ≡ (∃n ≤ ω)¬ϕ(n).

(ω is independent of parameters in ϕ)

∼ [(∃n ∈ ∗◆)ϕ(n)] ≡ (∀n ≤ ω)¬ϕ(n).

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Lost in translation

Hypernegation provides a translation between NSA and CRM:

Definition (Hypernegation)

∼ [(∃n ∈ ◆)ϕ(n)] ≡ (∀n ∈ ∗◆)¬ϕ(n). ∼ [(∀n ∈ ◆)ϕ(n)] ≡ (∃n ∈ ∗◆)¬ϕ(n). ∼ [(∀n ∈ ∗◆)ϕ(n)] ≡ (∃n ≤ ω)¬ϕ(n).

(ω is independent of parameters in ϕ)

∼ [(∃n ∈ ∗◆)ϕ(n)] ≡ (∀n ≤ ω)¬ϕ(n). With the hypernegation ∼, we get the usual results from CRM:

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Lost in translation

Hypernegation provides a translation between NSA and CRM:

Definition (Hypernegation)

∼ [(∃n ∈ ◆)ϕ(n)] ≡ (∀n ∈ ∗◆)¬ϕ(n). ∼ [(∀n ∈ ◆)ϕ(n)] ≡ (∃n ∈ ∗◆)¬ϕ(n). ∼ [(∀n ∈ ∗◆)ϕ(n)] ≡ (∃n ≤ ω)¬ϕ(n).

(ω is independent of parameters in ϕ)

∼ [(∃n ∈ ∗◆)ϕ(n)] ≡ (∀n ≤ ω)¬ϕ(n). With the hypernegation ∼, we get the usual results from CRM:

Theorem

In NSA, LPO is equivalent to MP plus LLPO

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Lost in translation

Hypernegation provides a translation between NSA and CRM:

Definition (Hypernegation)

∼ [(∃n ∈ ◆)ϕ(n)] ≡ (∀n ∈ ∗◆)¬ϕ(n). ∼ [(∀n ∈ ◆)ϕ(n)] ≡ (∃n ∈ ∗◆)¬ϕ(n). ∼ [(∀n ∈ ∗◆)ϕ(n)] ≡ (∃n ≤ ω)¬ϕ(n).

(ω is independent of parameters in ϕ)

∼ [(∃n ∈ ∗◆)ϕ(n)] ≡ (∀n ≤ ω)¬ϕ(n). With the hypernegation ∼, we get the usual results from CRM:

Theorem

In NSA, LPO is equivalent to MP plus LLPO

LPO: P ∨ ∼P, MP: ∼∼P → P, LLPO: ∼(P ∧ Q) → ∼P ∨ ∼Q (P, Q ∈ Σ1)

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Lost in translation

Theorem

In NSA, TFAE

1 LLPO

❘ ❘ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Lost in translation

Theorem

In NSA, TFAE

1 LLPO 2 (∀x ∈ ❘)(∼(x > 0) ∨ ∼ (x < 0))

❘ ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Lost in translation

Theorem

In NSA, TFAE

1 LLPO 2 (∀x ∈ ❘)(∼(x > 0) ∨ ∼ (x < 0)) 3 (∀x, y ∈ ❘)(xy = 0 → x = 0 ∨ y = 0)

◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Lost in translation

Theorem

In NSA, TFAE

1 LLPO 2 (∀x ∈ ❘)(∼(x > 0) ∨ ∼ (x < 0)) 3 (∀x, y ∈ ❘)(xy = 0 → x = 0 ∨ y = 0)

LLPO: ∼(P ∧ Q) → ∼P ∨ ∼Q (P, Q ∈ Σ1)

◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Lost in translation

Theorem

In NSA, TFAE

1 LLPO 2 (∀x ∈ ❘)(∼(x > 0) ∨ ∼ (x < 0)) 3 (∀x, y ∈ ❘)(xy = 0 → x = 0 ∨ y = 0)

LLPO: ∼(P ∧ Q) → ∼P ∨ ∼Q (P, Q ∈ Σ1)

Why does this connection exist? ◆

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Lost in translation

Theorem

In NSA, TFAE

1 LLPO 2 (∀x ∈ ❘)(∼(x > 0) ∨ ∼ (x < 0)) 3 (∀x, y ∈ ❘)(xy = 0 → x = 0 ∨ y = 0)

LLPO: ∼(P ∧ Q) → ∼P ∨ ∼Q (P, Q ∈ Σ1)

Why does this connection exist? Compare ◆ and N.

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

In classical NSA, a number is either finite or infinite. ◆❆ ◆

✶ ✷ ✸ ◆

❆ ✵ ✶ ✷ ✸

◆ ✵ ✶ ✶

✶

✷ ✷

❆

✷ ✶ ✸ ✶ ✷ ✸

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. ◆❆ ◆

✶ ✷ ✸ ◆

❆ ✵ ✶ ✷ ✸

◆ ✵ ✶ ✶

✶

✷ ✷

❆

✷ ✶ ✸ ✶ ✷ ✸

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆❆, an extension of ∗◆:

✲

0 1 . . . ω✶ . . . ω✷ . . . ω✸ . . .

◆

❆ ✵ ✶ ✷ ✸

◆ ✵ ✶ ✶

✶

✷ ✷

❆

✷ ✶ ✸ ✶ ✷ ✸

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆❆, an extension of ∗◆:

✲

0 1 . . . ω✶ . . . ω✷ . . . ω✸ . . .

◆

where ❆ = {✵, ✶, ✷, ✸, . . . }

◆ ✵ ✶ ✶

✶

✷ ✷

❆

✷ ✶ ✸ ✶ ✷ ✸

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆❆, an extension of ∗◆:

✲

0 1 . . . ω✶ . . . ω✷ . . . ω✸ . . .

◆

where ❆ = {✵, ✶, ✷, ✸, . . . }

◆ ✵ ✶ ✶

✶

✷ ✷

❆

✷ ✶ ✸ ✶ ✷ ✸

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆❆, an extension of ∗◆:

✲

0 1 . . . ω✶ . . . ω✷ . . . ω✸ . . . where ❆ = {✵, ✶, ✷, ✸, . . . }

◆, ✵-finite ✶ ✶

✶

✷ ✷

❆

✷ ✶ ✸ ✶ ✷ ✸

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆❆, an extension of ∗◆:

✲

0 1 . . . ω✶ . . . ω✷ . . . ω✸ . . . where ❆ = {✵, ✶, ✷, ✸, . . . }

◆, ✵-finite

• ✵-infinite

✶ ✶

✶

✷ ✷

❆

✷ ✶ ✸ ✶ ✷ ✸

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆❆, an extension of ∗◆:

✲

0 1 . . . ω✶ . . . ω✷ . . . ω✸ . . . where ❆ = {✵, ✶, ✷, ✸, . . . }

◆, finite

• infinite

✶ ✶

✶

✷ ✷

❆

✷ ✶ ✸ ✶ ✷ ✸

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆❆, an extension of ∗◆:

✲

0 1 . . . ω✶ . . . ω✷ . . . ω✸ . . . where ❆ = {✵, ✶, ✷, ✸, . . . }

◆, finite

• infinite

✶-finite

• ✶

✶

✷ ✷

❆

✷ ✶ ✸ ✶ ✷ ✸

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆❆, an extension of ∗◆:

✲

0 1 . . . ω✶ . . . ω✷ . . . ω✸ . . . where ❆ = {✵, ✶, ✷, ✸, . . . }

◆, finite

• infinite

✶-finite

• ✶-infinite
• ✶

✷ ✷

❆

✷ ✶ ✸ ✶ ✷ ✸

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆❆, an extension of ∗◆:

✲

0 1 . . . ω✶ . . . ω✷ . . . ω✸ . . . where ❆ = {✵, ✶, ✷, ✸, . . . }

◆, finite

• infinite

✶-finite

• ✶-infinite
• Fuzzy border:

✶

✷ ✷

❆

✷ ✶ ✸ ✶ ✷ ✸

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆❆, an extension of ∗◆:

✲

0 1 . . . ω✶ . . . ω✷ . . . ω✸ . . . where ❆ = {✵, ✶, ✷, ✸, . . . }

◆, finite

• infinite

✶-finite

• ✶-infinite
• Fuzzy border: no least ✶-infinite number

✷ ✷

❆

✷ ✶ ✸ ✶ ✷ ✸

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆❆, an extension of ∗◆:

✲

0 1 . . . ω✶ . . . ω✷ . . . ω✸ . . . where ❆ = {✵, ✶, ✷, ✸, . . . }

◆, finite

• infinite

✶-finite

• ✶-infinite
• ✷

✷

❆

✷ ✶ ✸ ✶ ✷ ✸

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆❆, an extension of ∗◆:

✲

0 1 . . . ω✶ . . . ω✷ . . . ω✸ . . . where ❆ = {✵, ✶, ✷, ✸, . . . }

◆, finite

• infinite

✶-finite

• ✶-infinite
• ✷

✷

❆

✷ ✶ ✸ ✶ ✷ ✸

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆❆, an extension of ∗◆:

✲

0 1 . . . ω✶ . . . ω✷ . . . ω✸ . . . where ❆ = {✵, ✶, ✷, ✸, . . . }

◆, finite

• infinite

✶-finite

• ✶-infinite
• ✷-finite
• ✷

❆

✷ ✶ ✸ ✶ ✷ ✸

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆❆, an extension of ∗◆:

✲

0 1 . . . ω✶ . . . ω✷ . . . ω✸ . . . where ❆ = {✵, ✶, ✷, ✸, . . . }

◆, finite

• infinite

✶-finite

• ✶-infinite
• ✷-finite
• ✷-infinite
• ❆

✷ ✶ ✸ ✶ ✷ ✸

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆❆, an extension of ∗◆:

✲

0 1 . . . ω✶ . . . ω✷ . . . ω✸ . . . where ❆ = {✵, ✶, ✷, ✸, . . . }

◆, finite

• infinite

✶-finite

• ✶-infinite
• ✷-finite
• ✷-infinite
• The infinite numbers are ‘stratified’ in |❆| many levels of infinity.

✷ ✶ ✸ ✶ ✷ ✸

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆❆, an extension of ∗◆:

✲

0 1 . . . ω✶ . . . ω✷ . . . ω✸ . . . where ❆ = {✵, ✶, ✷, ✸, . . . }

◆, finite

• infinite

✶-finite

• ✶-infinite
• ✷-finite
• ✷-infinite
• The infinite numbers are ‘stratified’ in |❆| many levels of infinity.

Then ω✷ is infinite ‘relative’ to ω✶

✸ ✶ ✷ ✸

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆❆, an extension of ∗◆:

✲

0 1 . . . ω✶ . . . ω✷ . . . ω✸ . . . where ❆ = {✵, ✶, ✷, ✸, . . . }

◆, finite

• infinite

✶-finite

• ✶-infinite
• ✷-finite
• ✷-infinite
• The infinite numbers are ‘stratified’ in |❆| many levels of infinity.

Then ω✷ is infinite ‘relative’ to ω✶ and finite ‘relative’ to ω✸.

✶ ✷ ✸

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

In classical NSA, a number is either finite or infinite. In ‘stratified’ or ‘relative’ NSA, there are ‘levels’ of infinity. Consider ◆❆, an extension of ∗◆:

✲

0 1 . . . ω✶ . . . ω✷ . . . ω✸ . . . where ❆ = {✵, ✶, ✷, ✸, . . . }

◆, finite

• infinite

✶-finite

• ✶-infinite
• ✷-finite
• ✷-infinite
• The infinite numbers are ‘stratified’ in |❆| many levels of infinity.

Then ω✷ is infinite ‘relative’ to ω✶ and finite ‘relative’ to ω✸. We write ω✶ ≪ ω✷ ≪ ω✸.

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Transfer

=‘All levels ❛ ∈ ❆ have the same properties as ◆.’ ◆❛ ◆❆ ❛ ❛ ❆ ◆ ◆ ◆ ◆❛ ◆❛ ◆❛

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Transfer

=‘All levels ❛ ∈ ❆ have the same properties as ◆.’ Define ◆❛ = {n ∈ ◆❆|n is ❛-finite}. ❛ ❆ ◆ ◆ ◆ ◆❛ ◆❛ ◆❛

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Transfer

=‘All levels ❛ ∈ ❆ have the same properties as ◆.’ Define ◆❛ = {n ∈ ◆❆|n is ❛-finite}.

Axiom schema (Πn)

For all ϕ ∈ ∆0 and ❛ ∈ ❆ (∀x1 ∈ ◆)(∃x2 ∈ ◆) . . . (Qxn ∈ ◆)ϕ(x1, . . . , xn) is equivalent to ◆❛ ◆❛ ◆❛

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Transfer

=‘All levels ❛ ∈ ❆ have the same properties as ◆.’ Define ◆❛ = {n ∈ ◆❆|n is ❛-finite}.

Axiom schema (Πn)

For all ϕ ∈ ∆0 and ❛ ∈ ❆ (∀x1 ∈ ◆)(∃x2 ∈ ◆) . . . (Qxn ∈ ◆)ϕ(x1, . . . , xn) is equivalent to (∀x1 ∈ ◆❛)(∃x2 ∈ ◆❛) . . . (Qxn ∈ ◆❛)ϕ(x1, . . . , xn)

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

Theorem (Πn-reduction)

Given Πn, for every ϕ ∈ ∆0, (∀x1 ∈ ◆)(∃x2 ∈ ◆) . . . (Qxn ∈ ◆)ϕ(x1, . . . , xn)

✶ ✷ ♥ ✶ ✷ ✸ ♥

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

Theorem (Πn-reduction)

Given Πn, for every ϕ ∈ ∆0, (∀x1 ∈ ◆)(∃x2 ∈ ◆) . . . (Qxn ∈ ◆)ϕ(x1, . . . , xn) is equivalent to (∀x1 ≤ ω✶)(∃x2 ≤ ω✷) . . . (Qxn ≤ ω♥)ϕ(x1, . . . , xn),

✶ ✷ ✸ ♥

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

Theorem (Πn-reduction)

Given Πn, for every ϕ ∈ ∆0, (∀x1 ∈ ◆)(∃x2 ∈ ◆) . . . (Qxn ∈ ◆)ϕ(x1, . . . , xn) is equivalent to (∀x1 ≤ ω✶)(∃x2 ≤ ω✷) . . . (Qxn ≤ ω♥)ϕ(x1, . . . , xn), (∗) with ω✶ ≪ ω✷ ≪ ω✸ ≪ . . . ≪ ω♥.

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

Theorem (Πn-reduction)

Given Πn, for every ϕ ∈ ∆0, (∀x1 ∈ ◆)(∃x2 ∈ ◆) . . . (Qxn ∈ ◆)ϕ(x1, . . . , xn) is equivalent to (∀x1 ≤ ω✶)(∃x2 ≤ ω✷) . . . (Qxn ≤ ω♥)ϕ(x1, . . . , xn), (∗) with ω✶ ≪ ω✷ ≪ ω✸ ≪ . . . ≪ ω♥. But (∗) is decidable!

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

Theorem (Πn-reduction)

Given Πn, for every ϕ ∈ ∆0, (∀x1 ∈ ◆)(∃x2 ∈ ◆) . . . (Qxn ∈ ◆)ϕ(x1, . . . , xn) is equivalent to (∀x1 ≤ ω✶)(∃x2 ≤ ω✷) . . . (Qxn ≤ ω♥)ϕ(x1, . . . , xn), (∗) with ω✶ ≪ ω✷ ≪ ω✸ ≪ . . . ≪ ω♥. But (∗) is decidable!

Theorem

A ≤T 0(n) ⇐ ⇒ A ∈ ∆n+1

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

The nonstandard Turing hierarchy

Theorem (Πn-reduction)

Given Πn, for every ϕ ∈ ∆0, (∀x1 ∈ ◆)(∃x2 ∈ ◆) . . . (Qxn ∈ ◆)ϕ(x1, . . . , xn) is equivalent to (∀x1 ≤ ω✶)(∃x2 ≤ ω✷) . . . (Qxn ≤ ω♥)ϕ(x1, . . . , xn), (∗) with ω✶ ≪ ω✷ ≪ ω✸ ≪ . . . ≪ ω♥. But (∗) is decidable!

Theorem

A ≤T 0(n) ⇐ ⇒ A ∈ ∆n+1 ⇐ ⇒ A ≤T Πn.

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Final Thoughts

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Final Thoughts

The two eyes of exact science are mathematics and logic, the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it sees better with one eye than with two.

Augustus De Morgan

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Final Thoughts

The two eyes of exact science are mathematics and logic, the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it sees better with one eye than with two.

Augustus De Morgan ...there are good reasons to believe that Nonstandard Analysis, in some version or other, will be the analysis of the future. Kurt G¨

• del

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Final Thoughts

The two eyes of exact science are mathematics and logic, the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it sees better with one eye than with two.

Augustus De Morgan ...there are good reasons to believe that Nonstandard Analysis, in some version or other, will be the analysis of the future. Kurt G¨

• del

Thank you for your attention!

Models of Computation Another bulwark of Computability The nonstandard Turing hierarchy

Final Thoughts

The two eyes of exact science are mathematics and logic, the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it sees better with one eye than with two.

Augustus De Morgan ...there are good reasons to believe that Nonstandard Analysis, in some version or other, will be the analysis of the future. Kurt G¨

• del

Thank you for your attention!

Any questions?