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 x : (1 + X + X N ) → X s : X → X l : X N → X Br : Set → Set Br X = 1 + X + X N Ω = µ Br x � X Br X Br ( ( [ . ] ) [ . ] ) � 0 , (+1) , sup � Br Ω 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 = l ( n �→ s n z ) ( [ ω ] ) X z s l 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 = D x (( [ α ] )( F X )( U x )) ���� 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 ( ω α ): 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 F ψ · F φ x �→ U ψ ( U φ x ) x �→ ( D φ x ) · D ψ ( U φ x ) How about ‘countable composition’ sup n ( φ n · φ n − 1 · · · · φ 0 ) ? Well, yes, it works. It is the basis for simulating the Veblen hierarchy χ α β . But it is a little heavy with subscripts, so let’s just look at φ ω = sup n ( 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 U n : ( Br X → X ) → (Br( F n X ) → F n X ). Now, eliding some of the more bureaucratic arguments, we have an inverse chain: D ... D ... D ... F 2 X . . . X F X Given ξ : F ′ X = ( � n : N ) F n X , form the ‘sup’ of ξ 0 = ξ , ξ 1 = ( n �→ D ( . . . ) ξ 0 ( n + 1)), ξ 2 = . . . using the sup at each level. (Rough) claim: if ( F , U , D ) simulates φ , which is normal, then the operation ξ �→ ξ ω maps F ′ X onto the inverse limit of the above chain, and simulates φ ω . Call this op C . Define U ′ . . . (giving a Br-algebra on F ′ X ) by applying/postcomposing C to ( U n ) 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 ω α . 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 ( F n , U n , D n ). 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 ε 0 . Then with one universe by iterating the large Br-algebra through these approximants, we can define approximants up to φ ε 0 0. And so on . . . with a tower of universes, up to Γ 0 . 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 super 2 universe, and so on. Peter Hancock (Nottingham) Church, simulation 15 March 2008, Swansea 8 / 8
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