Sets in HA K.O. Wilander Uppsala Universitet Background - - PowerPoint PPT Presentation

Sets in HA

ω

K.O. Wilander Uppsala Universitet

Background

• Extensional on top of Intensional
• PER construction, e.g. Hoffmann’s PER construction S0
• Preserve computations – lost if using relations
• Weaker background than ITT
• HAω – intuitionistic arithmetic (as presented in Troelstra & van Dalen)

HA

ω briefly

• Natural numbers
• Products
• Function spaces
• Combinators
• Minimal logic with equality
• Recursion operator and induction principle
• No proof objects

τ := ο | τ ! τ | τ ! τ S, K – typed & ∨ → ∀ ∃

Equality on objects

• Definitional equality is unavoidable
• All constructions should respect equality
• But we want extensional equality on maps!

Pullbacks A X B

f g

P

f ′ g ′

= =

P ′ !

Equality on objects

• Definitional equality is unavoidable
• All constructions should respect equality
• But we want extensional equality on maps!
• So must have a coarser equality on objects too

Partial Equivalence Relations in HA

ω

• PER A is: type σ together with a relation =A, symmetric and transitive
• Equality of structures on the type σ: A = B is

(∀xσ, yσ)(x =A y ↔ x =B y)

• Reflexive, symmetric, and transitive – seen easily
• Extend to relation on all PERs

Notation

• Aσ indicates A is a PER on type σ
• x ∈ A short for x =A x
• Bounded quantifiers: (∀x∈ Aσ)P(x) for (∀xσ)(x∈ A → P(x))

and (∃x∈ Aσ)P(x) for (∃xσ)(x∈ A & P(x))

Maps

• Maps Aσ ! Bτ are functions σ ! τ
• Equality: f = g means (∀xσ, yσ)(x =A y → f (x) =B g(y))
• Self-equality is extensionality
• PER of maps A ! B
• Respects equality of PERs
• Yields a category of PERs

Subsets and Quotients

• Subset relation Aσ ⊆ Bσ – then extend to partial order on all PERs
• Separation: {x∈ A : P(x)} has equality relation x =A y & P(x)
• Respects equality of PERs: A = B ⊆ C = D → A ⊆ D, etc.
• Subsets of A ⇔ extensional predicates up to equivalence
• So (externally) Heyting Algebra P( A )
• A ⊆ B – then inclusion map given by identity

Quotients

• Quotient A /R has equality relation x∈ A & y∈ A & xRy
• Relation R should be symmetric, transitive, and include equality
• Get quotient map A ! A /R given by the identity

Cartesian Product

• A PER structure on the product type
• Easy to see that the construction respects equality
• Set-theoretic language – similar to sets in Isabelle/HOL

Category PER and maps f : Aσ ! Bτ

• Injective: (∀x∈ A )(∀y∈ A )( f (x) =B f (y) → x =A y) – f monic
• Surjective: (∀y∈ B)(∃x∈ A )( f (x) =B y) – f epic
• Pseudo-split injective: (∃gτ!σ)(∀x∈ A )( g( f (x)) =A x) – f regular monic
• Pseudo-split surjective: (∃gτ!σ)(∀y∈ B)( g(y)∈ A & f ( g(y)) =B y) – f regular

epic, cover

• Pseudo-split – non-extensional inverse, a ‘choice operator’

Beware! The category is not balanced! (so not a pretopos)

Category PER and maps f : Aσ ! Bτ

• Injective: (∀x∈ A )(∀y∈ A )( f (x) =B f (y) → x =A y) – f monic
• Surjective: (∀y∈ B)(∃x∈ A )( f (x) =B y) – f epic
• Pseudo-split injective: (∃gτ!σ)(∀x∈ A )( g( f (x)) =A x) – f regular monic
• Pseudo-split surjective: (∃gτ!σ)(∀y∈ B)( g(y)∈ A & f ( g(y)) =B y) – f regular

epic, cover

• Pseudo-split – non-extensional inverse, a ‘choice operator’

Not balanced...

• Sufficient: not every injective surjection is pseudosplit – implies WEM
• Given proposition P

, consider the injective and surjective map {x∈N: (x = 0 & P) ∨ (x ! 0 & ~P)} ! {x∈N/(0=0): P ∨ ~P}

• A pseudosplitting f would then satisfy

P ∨ ~P → (f(0) = 0 & P) ∨ (f(0) ! 0 & ~P)

• But f(0) = 0 ∨ f(0) ! 0, and using this, get ~P ∨ ~~P

Category theoretical structure

• Cartesian Closed – nicely expressible in the set-theoretical language
• Particularly, the pullback is {x∈ A " B : f (#1(x)) =X g(#2(x))}
• Locally Cartesian Closed – Πf is harder to express
• All these constructions respect equality
• Also have NNO – N with standard equality

Images – for f : A ! B

• Two image constructions:
• {y∈ B : (∃ x∈ A )( f (x) =B y)} – least subset f factors through
• A / f – regular image factorisation

x ~ y if f (x) =B f (y)

Choice Principles

• At many levels:
• AC00 (∀xo)(∃yo)P(x,y) → (∃fo!o)(∀xo)P(x, f(x)) – base theory
• AC!AB (∀x∈ Aσ)(∃!y∈ Bτ)P(x,y) → (∃ f ∈ A ! B)(∀x∈ A )P(x, f(x)) – implies

balanced, so WEM

• (∀x∈ Aσ)(∃y∈ Bτ)P(x,y) → (∃fσ!τ)(∀x∈ A )P(x, f(x)) – implies epis regular,

WEM

• ACAB (∀x∈ A)(∃y∈ B)P(x,y) → (∃ f ∈ A !B)(∀x∈ A )P(x, f(x)) – implies

epis split, PEM

Countable & Dependent Choice

• AC0A (∀n∈ N)(∃x∈ A )P(n, x) → (∃f∈ N!A )(∀n∈ N)P(n, f(n)) – follows from

AC0σ in base theory

• DCA

(∀x∈ A )(∃y∈A )P(x,y)→(∀x∈A )(∃ f ∈N!A )( f (0) =A x & (∀n∈N)P( f n, f (sn))) – almost follows from DCσ in base theory but there’s a bounded quantifier – choice operator on A sufficient, but….

Real Numbers

• Standard option, construct Z and Q, and R as Cauchy sequences in Q
• Arithmetic +, -, " is easy
• Want: Inverse ⋅-1 : {x∈R: x # 0} ! R
• Fine but must have ‘slow’ Cauchy sequences
• But then Cauchy completeness is a problem (AC00 helps)

Real numbers, interval representation

• Define Z and Q+, two-point compactification of Q (±$ added)
• Then R as converging nested intervals
• Arithmetic – by interval arithmetic, including inverse
• There is now a function {α∈N!R: (∀i,j > n)(|αi - αj| < 2-n)} ! R taking

sequences to their limit, independent of proof!

Recap

• PER construction moved to arithmetic
• “Set theoretic language”
• LCCC, regular, all constructions given
• Choice difficult – but can add countable choice
• Sufficient for (some) analysis
• Also a version in predicative type theory

References

• Hofmann, Extensional Constructs in Intensional Type Theory
• Troelstra & van Dalen, Intuitionism in Mathematics
• Carlström, EM + Ext– + ACint is equivalent to ACext (MLQ 2004)
• Barthe, Capretta & Pons, Setoids in type theory (JFP 2003)