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

ω Sets in HA K.O. Wilander Uppsala Universitet

Background • Extensional on top of Intensional • PER construction, e.g. Hoffmann’s PER construction S 0 • Preserve computations – lost if using relations • Weaker background than ITT • HA ω – intuitionistic arithmetic (as presented in Troelstra & van Dalen)

ω briefly HA • Natural numbers τ := ο | τ ! τ | τ ! τ • Products • Function spaces S, K – typed • Combinators & ∨ → ∀ ∃ • Minimal logic with equality • Recursion operator and induction principle • No proof objects

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

Pullbacks P ′ ! P B = g g ′ f A X = f ′

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

ω Partial Equivalence Relations in HA • PER A is: type σ together with a relation = A , symmetric and transitive • Equality of structures on the type σ : A = B is ( ∀ x σ , y σ )(x = A y ↔ x = B y) • Reflexive, symmetric, and transitive – seen easily • Extend to relation on all PERs

Notation • A σ indicates A is a PER on type σ • x ∈ A short for x = A x • Bounded quantifiers: ( ∀ x ∈ A σ )P(x) for ( ∀ x σ )(x ∈ A → P(x)) and ( ∃ x ∈ A σ )P(x) for ( ∃ x σ )(x ∈ A & P(x))

Maps • Maps A σ ! B τ are functions σ ! τ • Equality: f = g means ( ∀ x σ , y σ )(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 A σ ⊆ B σ – 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 : A σ ! B τ • 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 Beware! The category is not balanced! • Pseudo-split injective: ( ∃ g τ ! σ )( ∀ x ∈ A )( g ( f (x)) = A x) – f regular monic (so not a pretopos) • Pseudo-split surjective: ( ∃ g τ ! σ )( ∀ y ∈ B )( g (y) ∈ A & f ( g (y)) = B y) – f regular epic, cover • Pseudo-split – non -extensional inverse, a ‘choice operator’

Category PER and maps f : A σ ! B τ • 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: ( ∃ g τ ! σ )( ∀ x ∈ A )( g ( f (x)) = A x) – f regular monic • Pseudo-split surjective: ( ∃ g τ ! 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 – Π f 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 x ~ y if f (x) = B f (y) • A / f – regular image factorisation

Choice Principles • At many levels: • AC 00 ( ∀ x o )( ∃ y o )P(x,y) → ( ∃ f o ! o )( ∀ x o )P(x, f(x)) – base theory • AC! AB ( ∀ x ∈ A σ )( ∃ !y ∈ B τ )P(x,y) → ( ∃ f ∈ A ! B )( ∀ x ∈ A )P(x, f(x)) – implies balanced, so WEM • ( ∀ x ∈ A σ )( ∃ y ∈ B τ )P(x,y) → ( ∃ f σ ! τ )( ∀ x ∈ A )P(x, f(x)) – implies epis regular, WEM • AC AB ( ∀ x ∈ A )( ∃ y ∈ B )P(x,y) → ( ∃ f ∈ A ! B )( ∀ x ∈ A )P(x, f(x)) – implies epis split, PEM

Countable & Dependent Choice • AC 0 A ( ∀ n ∈ N )( ∃ x ∈ A )P(n, x) → ( ∃ f ∈ N ! A )( ∀ n ∈ N )P(n, f(n)) – follows from AC 0 σ in base theory • DC A ( ∀ 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 DC σ 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 (AC 00 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)(| α i - α j | < 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 – + AC int is equivalent to AC ext (MLQ 2004) • Barthe, Capretta & Pons, Setoids in type theory (JFP 2003)