MHF 4102 - AX. SET THEORY FLORIDA INT'L UNIV.

TEST #2 - Spring 2009

Answer all 6 questions. Provide all rworking. An unjustified answer will receivBEGIN EACH QUESTION ON A SEPARATE PAGE.

TIME: 75 min.

asoning and show alllittle or no credit.

(20) 1.

(20) 2. Simplify the following ordinal ari4hmetic expressions asfar as possible.(a) 4+w+w2 (b) (w+2).(w+3) I (c) (w.2+1)2.

(15) 4 (a)

(b)

(15) 5. (a)

(b)

(15) 6. (a)

(b)

Gi ve the

operations:Prove that for

(i) K+11 = l1+K

definitions of thel following cardinalK+11, K.11, KJl.any two cardinals K ~nd 11,we always haveand (ii) K.11 = 11IK.

Define what are continuous and stri

J

tlY increasing class-functions from Q to Q. [Here Q = c ass of all ordinals]Let A be any set and P(A)= the power set of A. Prove thatA -< P (A) .

Write down what the Axiom of Choic

l

says.Prove in ZF that P (w), the powe set of w, can be

linearly ordered. (Hint: The set (;Ji well-ordered by E.)

(a) Define what is ordinal exponenti!tion and what is the

cofinality of a limit ordinal A.(b)

Prove that for any ordinals a, 'YJwe have (aO) y = aOy .[You may use any results you need about ordinal

arithmetic except this one, of co rse.]

(15) 3. (a) Define what A B means and write tlown what the Cantor-

Bernstein Theorem says.(b) Let Q = Set of all rational number; and N = Set of all

natural numbers. Prove that Q N.

- --

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