2.2
The Relevant Technology
In order to present Martins Axiom we must rst give some terminology.
Denition 2.9. A poset (for partially order set) is a pair P, where is a relation on P with
the following properties:
1. is reexive, ie if p P then p p.
2. is
Set Theory Course (F01): Lecture 3
Structure and Order
A. Polyadic Relations and Operations
1. N-tple (defined in terms of ordered couple). (Think about degenerate cases!)
2. N-ary relation and function. Special cases (binary etc.)
3. N-ary relation an op
Set Theory Course (F01): Lecture 2
The Basics of Sets, Relations and Functions
A. The Intuitive Notion of Set
1. One might understand set and the corresponding notion of memberhip in terms of the setbuilder.
2. The set-builder is a device for forming one
Short Guide to Further Reading on Set Theory
Let me emphasize that no reading beyond the Vaught is required. The following is merely
intended as a guide to those who would like to pursue the subject beyond its coverage in the course.
Of Historical Interes
Set Theory Course (F01): Lecture 1
A Short History of Set Theory
Cantor
Origin in the study of series; development of the basic concepts and theorems; the Continuum
Hypothesis.
Frege
Set-theory in its foundational role (as a basis for arithmetic and analy
Set Theory (F01): Exercises
Exercises on the Properties of Relations (Optional)
1. A relation is an equivalence if it is reflexive, transitive and symmetric.
(i) Show that these properties are independent, i.e. there is a relation having any two of these
Set Theory (F01): Assignment on well-orders
Starred exercises count towards base-line credit only.
*1. Which of the following is true and which false? Give reasons.
(i) if A = (A, #) is a well-order, then so is A* = (A, $);
(ii) if A = (A, #) is a well-or
Problem 1. Regard 1 as a model M = 1 , . Which axioms of ZFC does M satisfy?
Problem 2. Show that the following notions are absolute (ie, some formula dening them is
absolute):
1. x is a set of cardinality at most two.
2. x is an ordered pair.
3. x is a r
Problem 1. As in the proof that MA implies b = 2 , let P be the poset consisting of all pairs
p, A , where p Fn(, ) and A is a nite set of functions from into . We order P by saying
that q , B p, A if and only if p q, A B and for all n (dom(q ) \ dom(p) a
Problem 1. As usual, , and denote cardinals.
1. Show that ( ) = () .
2. Show that = 2 .
Problem 2. Let A be a given innite set. Show that the set of all bijections from A into A has
cardinality 2 .
Problem 3. Let an innite cardinal, let be another cardina
Set Theory (F01): Assignment
You may take for granted basic facts about natural numbers and cardinal numbers in doing these
exercises. Try to provide an intuitive proof even if you cannot state it rigorously.
1. Show that NN - 2N. (Hint: associate with ea
Set Theory (F01): Alternative Assignment
1. Provide detailed proofs of each of the following (i.e., specify the relevant function and
establish that it is a one-one correspondence of the required sort):
(i) L x A L;
(ii) cfw_a x A A, for any a;
(iii) A x