**Set theory as a foundattion for mathematics**

- A set is a collection of items
- Use set theory to build up all of mathematics
- Georg Cantor, Richard Dedekind 1870s
- Natural numbers can be “defined” as follows
- 0 corresponds to the empty set ∅
- 1 is the set {0, {0}} = {∅, {∅}}
- 2 is the set {1, {1}}
- . . .
- j + 1 is the set {j, {j}}
- Define arithmetic operations in terms of set building

**Russell’s Paradox**

- Set theory assumes the emptyset ∅ and basic set building operations
- Union ∪, Intersection ∩, Cartesian product ×, . . .
- Set comprehension — subset that satisfies a condition
- Is every collection a set? Is there a set of all sets?
- Consider S, all sets that do not contain themselves
- S is a set, by set comprehension
- Does S belong to S?
- Yes? But elements of S do not contain themselves
- No? Any set that does not contain itself should be in S
- Russell’s Paradox — also discovered by Ernst Zermelo

Cannot have “set of all sets”

**Sets and collections**

- Russell’s Paradox tells us that not every collection can be called a set
- Collection that is not a set is sometimes called a class
- The paradox had a major impact on set theory as a logical foundation of mathematics
- For us, just be sure that we always build new sets from existing sets
- Don’t manufacture sets “out of thin air” — “set” of all sets

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