MHF 4102 - AXIOMATIC SET THEORY FLORIDA INT'L UNIV.
Review for Test #2 (Jan. 6th)                                                                       FALL  2009

       REMEMBER TO BRING AN 8’’x11”  BLUE EXAM BOOKLET FOR THE TEST

               KEY CONCEPTS AND MAIN DEFINITIONS:

       Ordinal addition, Ordinal multiplication, Ordinal exponentitaion, Properties of the Cumula-
       tive Hierarchy, Cantor's Normal Form, weak & strong Goodstein sequences, Continuous   
       ordinal functions, increasing ordinal functions, Normal functions,  Fixed-points of a function,
       Equipotent (equipollent) sets, The sizes of N,  Z,  A,  R,  C,  NxN,  F(N,2),  P(N), P_FIN(N),
       SEQ(N), SEQ_FIN(N),  and F(N,N);  Initial ordinals (Cardinals), The Hartog number of a set,
       The alephs,  Cardinal addition & multiplication, Cardinal exponentiation, Axiom of Choice,
       (AC), Well Ordering Principle (WOP), Zorn's Lemma, Contniuum Hypothesis (CH),
       Generalized Contniuum Hypothesis (GCH), The cofinality of a limit ordinal, Singular & Regular
       cardinals, Weakly & Strongly inacessible cardinals,  Non-well-founded sets (hypersets),
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       The Transfinite Partition Algorithm, The minimal representation of a hyperset.

              MAIN PROBLEM SOLVING TECHNIQUES:

1. Proving various statements about ordinals by using transfinite induction. and
various statements about the cumulative hierarchy by using transfinite induction
       2. Simplifying expressions involving Ordinal arithmetic.

3.   Expressing ordinal numbers in various bases (base 2, base omega)
4. Finding the least fixed-point and other fixed-pointsof various normal functions
5. Proving cardinality results by using injections & bijections
6. Proving various results by using Cantor's Diagonal Argument
       7. Proving various results by using the Cantor--Bernstein & Squeezing Theorems.

       8. Proving results by using the Axiom of Choice & by using the Well Ordering Principle.
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       9.   Finding the minimal representation of a hyperset.
       10. Determining if two hypersets are equal.

               MAIN THEOREMS

       1. The Fixed-point Theorem for Normal functions
2. The Cantor Diagonal Theorem
       3. The Cantor-Bernstein Theorem
4. If  K & M are infinite cardinals, then K+M = K.M = max{K,M}.
5. If 1 < M < K⁺,  then M^K = 2^K for all infinite cardinals K.

6.   The Axiom of Choice is logically equivalent to the Well Ordering Principle.
       7.    For any ordinal a,  cof(2^{alepf sub a})  >  aleph sub a.