A Book of Set Theory by Charles C. PinterThis 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 Axioms are not Universal statements of truth but rather what we which to use as premises of statements and deductions are Independant of meaning. They are mechanical, following rules.
We need predicates, logical connectives & quantifiers