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CAPITA SELECTA: MODEL THEORY, AXIOMATIC SET THEORY

TOPIC FOR 2009: DESCRIPTIVE SET THEORY

ASSIGNMENT 4 (DUE 9TH OF NOVEMBER 2009)

(a) Show that a function is 01 iff it is computable. Hint: From a 01 and a

01 definition of f

you can define f recursively. The other direction is similar (but simpler) to the theoremthat all Borel sets are 11.

(b) Show that the inverse image of a 01 set through a computably coded continuous functionis 01. Hint: Note that the intersection of a computable sequence of

01 classes (as the

sets of paths through computable trees) is 01. So prove that this inverse image is theintersection of the classes of paths through a computable sequence of trees. You couldalso work with closed sets, instead of using trees.

(c) Show that there are uncountably many well founded trees.

(d) Show that the height/length of a well founded tree is a countable ordinal.

(e) Show that given a countable ordinal , there are countably many well founded trees ofheight . So WF is countable. Hint: By induction on .

(f) Prove the normal form theorem for 11 predicates in the space . That is, for every 11

predicate P there is a computable sequence of (computable) trees Tn such that(1) P (n) Tn WF Tn WFG.

Also prove the corresponding normal form for 11 (as above, only without computable).Hint: Suppose that P is 11. Then

n A mP (,m, n)for a computable predicate P . Without loss of generality we can assume that if P (,m, n)then P (, i, n) for all i m. We need to make a suitable sequence of trees Tn as above,using P as ingredient. We just need to read-off P as a tree. We let Tn be the set of mfor all and m such that P (,m, n). Verify that this set of string is a (computable)tree, for each n. Then verify (1). A similar argument works for 11.

(g) Let11 = sup{||T || | T WF and T is 11}.

Show that 11 = CK1 .

Hint: It suffices (why?) to show that

11 sup{||T || | T WF and T is computable}.Assume for a contradiction that some well-founded 11 tree F has height which is greaterthan all computable ordinals. By the 11 representation theorem (or, normal form theorem)every 11 relation P can be written as

P (n) Tn WFwhere Tn is a computable function from to the space of computable trees. But then wehave

P (n) Tn WF ||Tn|| ||F ||.Explain that this gives a 11 definition of P . The result follows since

11 6 11.

1

2 ASSIGNMENT 4 (DUE 9TH OF NOVEMBER 2009)

(h) Show that the class 11 is the smallest class which contains the computable sets andis closed under effective unions and complements. Hint: First show that 11 has thementioned properties. Then by induction on the computable ordinals show that everyclass M with these closure properties contains 11. Instead of the , classes of thehyperarithmetic hierarchy you can use the fact that 11 contains the sets which havecomputable Borel codes. Show that for every computable ordinal the sets which haveBorel code with ||T|| = belong to M.

(i) Show that every 11 set is the image of a 01 set through a 11 computably coded continuous

function from N to N . Hint: Follow the corresponding boldface proof and replace thenotions with their effective counterparts.

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