A Book of Set Theory by Charles C. Pinter
This is a rather interesting book that I am slowly making my way through. The ideas contained in the book, the authors tone, and care of writing are remarkable. What comes through is a clear demonstration of how much the Author cares about Mathematics, and this makes you care too.
Notes, Ideas & Links
- Georg Cantor
- Virtual & Actual Infinites Actual: Many objects are conceived as existing simultaneously Virtual: The Potential to exceed any finite quantity
- Mathematicians dreamed of a “Unifying branch of mathematics "
- The Ancient schools thought this unification would lie in geometry
- 19th Century Mathematicians such as Weierstrass & Dedekind thought unification could be achieved under the rubric of Arithmetic.
- It was shown that every Real Number could be represented as a sequence of rational numbers. And Rationals as pairs of Integers.
- We can in turn show how Natural numbers can be represented by sets
- Set Theory Paradoxes, there are two main types:
- Logical - Russell’s Paradox
- Semantic - Berry’s Paradox
- The paradoxes arise because in Naive Set Theory we take the position that we can describe properties of an object and the decribe sets of objects based on these properties.
- However certain properties lead to paradoxes
- So we must ask, what are legitimate properties? How do we decribe the existence of sets.
- There have been three major schools for solving these problems:
- Axiomatic
- predicates, logical connectives & quantifiers
- Logistic
- Intuitionist