Models of set theory

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(1) Show that the following are oquivalent:

(a) y is an inaccessible cardinal in L(x), for every x  w.

(b) For every x  w, the of L(r) is a countable ordinal.

(2) Let A be any set. Show that L(A) satisfies AC if and only if L(A) contains a well-ordering of TC({A)). For the 'if' direetion, show that if <A is a well-ordering of TC({A}) in L(A), then in L(A) there is a global well-ordering that is E-definable using <A as a parameter.

(3) Show that L(P(ω)) = L(Vw+1) = L(Hw1).

(4) Show that every set in L(P()) is definable in L(P(w)) with for every AE L(P()) there is a formmla (r), possibly with a formula with subsets of o and ordinals as parameters. i.e.,d subsets of w and ordinals as parameters, such that

L(P(ω)) - x(x E Α φ(z)).

(5) A map e: PQ, where P and Q are partial orderings,is called a projection embedding if it preserves the orderingd relation and the preimage e-D] of any dense open subset D of Q is predense in P, mcaning that every condition in P is compatible with some clement of e-'D].

Let M be a countable transitive model of ZFC. Show that, in M, for any posets P and Q, an embedding e:P→Q is a projection embedding iff, writing H for the standard name ford the P-generic filter, P forces that the set G:= {gEQ: ЭpE H (e(p) ≤ q)) is a Q-generic filter over M. 

(6) Let A be a subset of {d: d is infinite} with thed finite intersection property, i.e., the intersection of every finite subset of A is infinite.

Find a cee partial ordering P such that forcing with Padds some c which is almost contained in all elements of A,i.e., such that e d is finite for all d E A.

(7) Let M be a countable transitive model of ZFC*+V = L. Let c Cw be a Cohen real over M (i.e., the characteristic function of c is UG, for some G generic over M for the Cohen poset, namely the poset whose conditions are finite sequences of O'sand 1's, ordered by 2). Find a forcing extension M[c][G] of M[c] in which c is ordinal-definable.  Conclude that

M[e][G]= "ZFC + L ≠ HOD". 

(Hint: Force that 2» = Nn+1 if and only if n E c.)

(8) Consider the w-product of Cohen forcing C, with full support.  So, conditions are w-sequences of finite sequences of natural numbers, ordered by coordinate-wise reversed inclusion. Show that Cw is not ccc.  Show that, in fact, forcing withC collapses the reals.  i.e., in any generic extension, the set of reals of the ground model (and therefore also wi) is countable.