Church encodings of ordinals, and simulation of ordinal functions - - PowerPoint PPT Presentation

Church encodings of ordinals, and simulation of ordinal functions

Peter Hancock hancock@spamcop.net

Nottingham University CSIT

15 March 2008, Swansea

Peter Hancock (Nottingham) Church, simulation 15 March 2008, Swansea 1 / 8

Ordinals as iterators

“A number is the exponent of an operation.” (T, 6.021) X : Set z : X s : X → X l : X N → X    x : (1 + X + X N) → X Br : Set → Set Br X = 1 + X + X N Ω = µ Br Br X

x

X

Br Ω

0,(+1),sup

• Br (

[.] )

• Ω

( [.] )

• Peter Hancock (Nottingham)

Church, simulation 15 March 2008, Swansea 2 / 8

AMEN

( [α + β] ) X z s l = ( [β] ) X (( [α] ) X z s l) s l ( [α × β] ) X z s l = ( [β] ) X z, (x → ( [α X x s l] )) s l ( [α ↑ β] ) X z s l = ( [β] ) (X → X) s (f , x → ( [α] ) X x f l) (g, x → l(n → g n x)) z ( [0] ) X z s l = z ( [ω] ) X z s l = l(n → sn z)

Peter Hancock (Nottingham) Church, simulation 15 March 2008, Swansea 3 / 8

Algebra

Modulo βη, (0, +) a monoid. (1, ×) a monoid. α × 0 = 0, α × (β + γ) = α × β + α × γ α ↑ 0 = 1, α ↑ (β + γ) = α ↑ β × α ↑ γ α ↑ 1 = α, α ↑ (β × γ) = (α ↑ β) ↑ γ In particular, α + 0 = α α + (β + 1) = (α + β) + 1 α × 0 = 0 α × (β + 1) = (α × β) + α α ↑ 0 = 1 α ↑ (β + 1) = (α ↑ β) × α So, our definitions are correct.

Peter Hancock (Nottingham) Church, simulation 15 March 2008, Swansea 4 / 8

Simulation

( [φ α] ) X z s l

• x

= D x(( [α] )(F X)(U x)) where F : Set → Set U : (Br X → X) → (Br(F X) → F X): ‘uplifts’ a Br-algebra on carrier X to another on F X. D : (Br X → X) → F X → X: ‘drops’ from F X to X. Example (ωα): F X = X → X, U x = s, (f , x → l(n → f n x)), (g, x → l(n → g n x)), D x = (f → f z) .

Peter Hancock (Nottingham) Church, simulation 15 March 2008, Swansea 5 / 8

Some nice closure properties

Closed under composition. φ · ψ simulated by Fψ · Fφ x → Uψ(Uφx) x → (Dφ x) · Dψ (Uφ x) How about ‘countable composition’ supn(φn · φn−1 · · · · φ0) ? Well, yes, it works. It is the basis for simulating the Veblen hierarchy χα

β.

But it is a little heavy with subscripts, so let’s just look at φω = supn(φ · φ · · · · φ

• n

).

Peter Hancock (Nottingham) Church, simulation 15 March 2008, Swansea 6 / 8

Sup of a sequence

Given F : Set → Set, form F ′ X = ( n : N) F n X. Given U : (Br X → X) → (Br(F X) → F X), form Un : (Br X → X) → (Br(F n X) → F n X). Now, eliding some of the more bureaucratic arguments, we have an inverse chain: X F X

D...

• F 2 X

D...

• . . .

D...

• Given ξ : F ′ X = ( n : N) F n X, form the ‘sup’ of ξ0 = ξ,

ξ1 = (n → D(. . .)ξ0(n + 1)), ξ2 = . . . using the sup at each level. (Rough) claim: if (F, U, D) simulates φ, which is normal, then the

• peration ξ → ξω maps F ′ X onto the inverse limit of the above

chain, and simulates φω. Call this op C. Define U′ . . . (giving a Br-algebra on F ′ X) by applying/postcomposing C to (Un) above.

Peter Hancock (Nottingham) Church, simulation 15 March 2008, Swansea 7 / 8

The (F, U, D)’s form a large Br-algebra

The zero: take F X = X → X, . . ., that simulates ωα. The successor: the operation that takes (F, U, D) to X → ( n : N) F n X, . . . as on the previous slide. (More or less, takes us from a normal function φ to its Veblen derivative. The limit: we have an ω-sequence of (Fn, Un, Dn). The idea is quite similar to what we do in the successor case, except the steps in the chain are heterogeneous. With no universes, we can define approximants up to ε0. Then with one universe by iterating the large Br-algebra through these approximants, we can define approximants up to φε00. And so on . . . with a tower of universes, up to Γ0. Rash claim: I expect that the same techniques (with essentially no new ideas), can be used to obtain similar (lower bounds) results for a superuniverse, a super2 universe, and so on.

Peter Hancock (Nottingham) Church, simulation 15 March 2008, Swansea 8 / 8