This document discusses sets and axioms of set theory. It provides examples of sets such as a bag of potato chips or a university. It then discusses the axioms of set theory, including the axiom of extension, null set, pairing, union, power set, separation, and infinity. It provides definitions of well-ordered sets and discusses Zorn's lemma and its proof. The document also discusses concepts like maximal elements, linear orderings, and the axiom of choice.

2. Sets are all around us
-For instance, a bag of potato chips is a set containing certain number of individual
chip’s that are its elements.
-University is another example of a set with students as its elements. By elements, we
mean members.
But sets should not be confused as to what they really are:
-A daughter of a blacksmith is an element of a set that contains her mother, father, and
her siblings, and even other families that live in the nearby town. So a set itself can be an
element of a bigger set.

3. -In mathematics, axiom is defined to be a rule or a statement that is
accepted to be true regardless of having to prove it. In a sense, axioms are
self evident.
-In set theory, we deal with sets. Each time we state an axiom, we will do so
by considering sets.
Example,the set containing the blacksmith family might make it seem as if
sets are finite. In truth, they are not! The set containing all the natural
numbers {1, 2, 3, ···} is an infinite set.

4. • Axiom 1 (the axiom of extension)
• Axiom 2 (the axiom of the null set)
• Axiom 3 (the axiom of pairing)
• Axiom 4 (the axiom of union)
• Axiom 5 (the axiom of the power set)
• Axiom 6 (the axiom of separation)
• Axiom 7 (the axiom of replacement)
• Axiom 8 (the axiom of infinity)
• Axiom 9 (the axiom of regularity)

5. • Given any nonempty set Y whose members are pairwise
disjoint sets, there exists a set X consisting of exactly one
element taken from each set belonging to Y. (Lay 94)
• Let {Xα } be a family of nonempty sets. Then there is a set X
which contains, from each set Xα , exactly one element.
(Garrity 207)

6. 1924, S. Banach and A. Tarski 1939, Kurt Gödel Early 1960s, Paul Cohen

7. When we have a finite number of sets?
let X1={a,b} and X2={c,d}. let X={a,c}.
When we have an infinite number of sets whose elements are well-
ordered?
well-ordering of the natural numbers
When we have an infinite number of sets whose elements are
not well-ordered?

9. • We can also say that all sets can be well-
ordered.
• "The Axiom of Choice gives no method for
finding the set X; it just mandates the existence
of X". (Garrity 208)

10. A total order ≤ on a set E is said to be a well-order on E provided that, for each A a subset
of E, there exists an m an element of A such that m ≥b for each b an element of A. The
element m is said to be the maximal element of A (on E with respect to ≤).
Given S a subset of K, we say that q an element of K is a ≤-upper bound of S provided
that s≤ q for each s in S.
A relation ~ on a set X is a partial ordering of X if it is transitive (if x~y and y~z implies x~z)
and antisymmetric (x~y and y~x implies x=y)
For example, ≤ is a partial ordering of the real numbers.

11. A partial ordering ~ on a set X is a linear ordering on X if for any two
elements x, y in X, either x~y or y~x.
Again, the relation ≤ is a linear ordering on the real numbers.
A linearly ordered subset E of X is maximal if any linearly ordered
subset of X is contained in E.

12. • The well-ordering principle
Given any set A, there exists a well-order in A.
Recall:
A total order ≤ on a set E is said to be a well-order on E provided
that, for each A a subset of E, there exists an m an element of A such
that m ≥b for each b an element of A.
• Zorn’s Lemma
Let X be a partially ordered set such that every linearly ordered subset has
an upper bound. Then X has a maximal element.

14. Zorn’s Lemma. Let X be a partially ordered set such that every linearly
ordered subset has an upper bound. Then X has a maximal element.
Proof:
Let M be the maximal linearly ordered set claimed by the maximal principle,
which states that every partially ordered set contains a maximal linearly ordered
subset. An upper bound for M is a maximal element of X.
Definition: Let X be a set partially ordered by the relation ~ and let E be a subset
of X. An upper bound of a subset E of X is an element x of X such that y~x for all
y in E. If x is an element of E, then x is a maximal element of E.

15. Corollary of the Axiom of Choice. Let X be a set. There exists a function f: 2X → X such
that f (E) is an element of E for every E a subset of X. That is, one may choose an
element out of every subset of X.
Proof:
Let f: 2 X → X be a function, as in corollary above, whose existence is guaranteed by the
Axiom of Choice. Set x1 = f (X) and xn= f (X – (union of xj for j=1 to j=1-n for n ≥2))
The sequence of {xn } can be given the ordering of the natural numbers and, as such, is
well-ordered. A well-ordering for is constructed by rendering transfinite such a process.

16. Let D be a subset of X and let ~ be a linear ordering defined on D. A subset E of D is a segment relative
to ~ if for any x an element of E, all y elements of D such that y ~ x belong to E.
The segments of {xn} relative to the ordering induced by the natural numbers are the sets of the form
{x1 , x2 , … , xm} for some m in the natural numbers. The union and intersection of two segments is a
segment. The empty set is a segment relative to any linear ordering ~.
Denote by F the family of linear orderings ~ defined on subsets D of X and satisfying the following:
If E as subset of D is a segment, then the first element of (D – E) is f (X – E).
Such a family is not empty since the ordering of the natural numbers on the domain D = {xn } is in F.

19. • KURATOWSKI'S LEMMA
Theorem
Formulation 1
Let (S,⪯),S≠∅ be a non-empty ordered set.
Then every chain in S is the subset of some maximal chain.
Formulation 2
Let S be a set of sets which is closed under chain unions.
Then every element of S is a subset of a maximal element of S under the
subset relation.

20. • HAUSDORFF'S MAXIMAL PRINCIPLE
Theorem
Formulation 1
Let (P,⪯) be a non-empty partially ordered set.
Then there exists a maximal chain in P.
Formulation 2
Let A be a non-empty set of sets.
Let S be the set of all chain of sets of A (ordered under the subset relation).
Then every element of S is a subset of a maximal element of S under the
subset relation.

21. • TUKEY'S LEMMA
Theorem
Formulation 1
Let S be a non-empty set of finite character.
Then S has an element which is maximal with respect to the subset relation.
Formulation 2
Let S be a non-empty set of finite character.
Then every element of S is a subset of a maximal element of S under the
subset relation.

22. • TYCHONOFF'S THEOREM

23. • KELLEY'S THEOREM
Theorem
Let (D,⪯) be a directed set,
Let S be a non-empty set.
Let n:D→S be a net in S.
Then n has a universal subnet.

24. YOU!