15.Alternative Set Theories

A Brief Introduction to Set Theory with Classes

(Godel-Bernays "GB")

For reference here, again, is pre-Russel's paradox Set Theory.

Axiom 1: Two sets are the same if and only if (write "iff") they have the same members.Symbolically,

$\QTR{Large}{X=Y}$

Axiom 2: Let be a proposition about mathematical objects then true $\QTR{Large}{\}}$ is a Class.

15.1 Terms and Relations in GB:

- read " $\QTR{Large}{X}$ is a Set"
$\QTR{Large}{Cl(X)}$ - read " $\QTR{Large}{X\ }$ is a Class"
- read " $\QTR{Large}{X\ }$ is a member of $\QTR{Large}{Y\ }$ "

15.2 Some Axioms:

- Everything is either a Set or a Class.
- Every Set is a Class.
- Every member of a Set or a Class is a Set.
Let be a proposition about mathematical objects thentrue $\QTR{Large}{)\}}$ is a Class.

Restated:

Let be a proposition about mathematical objects then
other ZF- type axioms.

15.3 Russel's paradox revisited:

Rewriting 4. with , the Russel example now reads. The exists a Class $\QTR{Large}{Y}$ consisting of all Sets that do not contain themselves as members.

One asks is

If then a contadiction.
If then if and only if . Hence if $\QTR{Large}{Y}$ is a Class but not a Set we do not have a contradiction. "The Barber does not live on the island."

Contructive Set Theory

A CONSTRUCTIVE VERSION OF CANTOR'S THEOREM

Classical Set Theory: is uncountable .

Constructive Set Theory: Let be a sequence of real numbers. let and be real numbers with $\QTR{Large}{<}$ . Then there exists a real number $\QTR{Large}{x}$ such that

$\$ and $\QTR{Large}{x}$
for all $\QTR{Large}{n}$ .

15.4 Definitions:

A real number $\QTR{Large}{b}$ is called a constructive supremum, or constructive least upper bound, of if it is an upper bound for $\QTR{Large}{A}$ and if for each there exists with . We write
A real number $\QTR{Large}{l}$ is called a contructive infimum, or constructive greatest lower bound, of A if it is a lower bound for and if for each each there exists with . We write

15.4 Theorem(Contructive lub). Let A be a nonempty set of real numbers that is bounded above. Then exists if and only if for all with $\QTR{Large}{x<y}$ , either $\QTR{Large}{y}$ is an upper bound for $\QTR{Large}{A}$ or there exists with $\QTR{Large}{x<a}$ .